On the dark matter haloes inner structure and galaxy morphology

Abstract

In the present paper, we extend the study of Del Popolo (Mon. Not. R. Astron. Soc. 408:1808, 2010) to determine the slope of the inner density profile of galaxy haloes with different morphologies.

We study how galaxy morphology changes the relation between the inner slope of the galaxy halo density profile, $\alpha$, and the stellar mass, $M_{*}$, or rotation velocity, $V_{\mathrm{rot}}$. For this, we use the model of Del Popolo (Astrophys. J. 698:293, 2009) in combination with observed data from the Romanowsky and Fall (Astrophys. J. Suppl. Ser. 203:17, 2012) sample of elliptical and spiral galaxies, the Local Group sample compiled by McConnachie (Astrophys. J. 144:4, 2012), and the simulation results by Cloet-Osselaer et al. (Mon. Not. R. Astron. Soc. 442:2909, 2014).

We find that the slope $\alpha$ flattens monotonically, from $\alpha \simeq -1$ at $V_{\mathrm{rot}} \simeq 250~\mbox{km/s}$, to $\alpha \simeq 0$. After $V_{\mathrm{rot}}\simeq 25~\mbox{km/s}$ the slope starts to steepen. The steepening happens in the mass range dominated by non-rotationally supported galaxies (e.g., dSphs) and depends on the level of offset in the angular momentum of rotationally and non-rotationally dominated galaxies. The steepening is a consequence of the decrease in baryons content, and angular momentum in spheroidal dwarf galaxies.

We finally compare our result to the SPH simulations of Di Cintio. Our result is in qualitative agreement with their simulations, with the main difference that the inner slope $\alpha$ at small stellar masses ($M_{*} \lesssim10^{8} M_{\odot}$) is flatter than that in their simulations.

As a result, the claim that finding a core in dwarf galaxies with masses slightly smaller than $\simeq 10^{6} M_{\odot}$ (as in the Di Cintio, or Governato, supernovae feedback mechanism) would be a problem for the $\Lambda\mbox{CDM}$ model which should probably be revised.

Appendices

Appendix A: Model

The model used to determine the $\alpha$–$M_{*}$ ($\alpha$–$V_{\mathrm{rot}}$) relation was introduced in DP09 and Del Popolo and Kroupa (2009), and then applied in several other papers to study the universality of the density profiles (Del Popolo 2010, 2011), the density profiles in galaxies (Del Popolo 2012a; Del Popolo and Hiotelis 2014) and clusters (Del Popolo 2012b, 2014b), and the inner surface-density of galaxies (Del Popolo et al. 2013). The model is semi-analytical and includes an improved secondary infall model (SIM) (e.g., Gunn and Gott 1972; Hoffman and Shaham 1985; Del Popolo and Gambera 1997; Williams et al. 2004; Ascasibar et al. 2004; Hiotelis and Del Popolo 2006, 2013; Cardone et al. 2011a, 2011b; Del Popolo et al. 2013a, 2013b, 2013c). The latter takes into account random and ordered AM effects (Ryden and Gunn 1987; Ryden 1988), exchange of energy and AM from baryons to DM through DF (e.g., El-Zant et al. 2001, 2004), adiabatic contraction of DM (Blumenthal et al. 1986; Gnedin et al. 2004; Gustafsson et al. 2006). The model was further improved to take into account cooling, star formation, supernovae feedback, and reionization (e.g., Del Popolo and Hiotelis 2014) (see the following).

The model can be summarized as follows. Initially the proto-structure is in the linear phase, containing DM and diffuse gas. The proto-structure expands to a maximum radius then re-collapses, first in the DM component that forms the potential well in which baryons will fall. Baryons subject to radiative processes form clumps, which collapse to the center of the halo while forming stars. In the collapse phase, baryons are compressed (adiabatic contraction) (at $z \simeq 5$ in the case of a $10^{9} M_{\odot}$ galaxy (DP09)) so making the DM profile more cuspy. The clumps collapse to the galactic center because of dynamical friction (DF) between baryons and DM, transferring energy and AM to the DM component (El-Zant et al. 2001, 2004), increasing its random motion and producing a predominant motion of DM particles outwards, reducing the central density.¹⁷ The cusp is heated, and a core forms. This is the main mechanism of core formation in dwarf spheroidals and spirals.¹⁸

In the case of spiral galaxies, the effect of the previous mechanism is amplified by the larger AM acquired by the proto-structure through tidal torques (ordered AM), and by random AM, with respect to spheroids. Supernovae feedback acts similarly to the mechanism now discussed (see Pontzen and Governato 2014) contributing to the core formation (see Madau et al. 2014).

In the following stage, supernovae explosions produce expulsion of gas in different events, clearly connected to the explosion of each supernova, leaving a lower stellar density with respect to the beginning. Feedback destroys the smallest clumps soon after a small part of their mass is transformed into stars (see Nipoti and Binney 2015).

1.1 A.1 Density profile formation

The model follows the evolution of a perturbation starting from the linear phase, expanding with the Hubble flow, from initial radius $x_{i}$, to the phase of maximum expansion $x_{\mathrm{m}}$ (named apapsis or turn-around radius $x_{\mathrm{ta}}$) (Peebles 1980).

The final density profile is given by (e.g., Fillmore and Goldreich 1984; Hiotelis 2002)

$$ \rho(x)=\frac{\rho_{\mathrm{ta}}(x_{\mathrm{m}})}{f^{3}} \biggl[1+\frac{d \ln f}{d \ln g} \biggr]^{-1}, $$

(15)

where $f(x_{i})$ is the collapse factor of a shell, namely the ratio between the final radius and the turn-around radius, and, as shown in DP09, depends on the radial velocity of the shell. This radial velocity can be obtained by integrating the equation of motion of the shell:

$$ \frac{dv_{r}}{dt}=\frac{h^{2}(r,\nu)+j_{\mathrm{r}}^{2}(r,\nu)}{r^{3}}-G(r)-\mu\frac{dr}{dt}+ \frac{\Lambda}{3}r, $$

(16)

where the acceleration $G(r)= GM_{\mathrm{T}}/r^{2}$, $\Lambda$ is the cosmological constant, $\mu$ the coefficient of dynamical friction, $h(r,\nu )$ ¹⁹ is the ordered SAM generated by tidal torques (Hoyle 1949; Peebles 1969; White 1984; Ryden 1988; Eisenstein and Loeb 1995; Catelan and Theuns 1996), and $j_{r}(r,\nu)$ is the random AM (see Ryden and Gunn 1987 and the following of the present paper).

The “ordered angular momentum” is calculated by obtaining the root-mean-square (rms) torque, $\tau(r)$, on a mass shell and then calculating the total SAM, $h(r, \nu)$, acquired during expansion by integrating the torque over time (Ryden 1988, Eq. (35)) (see Sect. C2 of DP09). As shown in DP09, in the case of a galaxy like the Milky Way, the AM obtained is $2.5 \times 10^{74}~\mbox{g}\,\mbox{cm}^{2}/\mbox{s}$ (Appendix C1, DP09).

“Random angular momentum” (see RG87; DP09, Appendix C2) is taken into account assigning an SAM at turnaround (see Appendix C2 of DP09 for details).

Equation (16) shows that the larger the content of AM of a system, the larger the collapse time of particles at a distance $r$.²⁰

Angular momentum sets the shape of the density profile in the inner regions. For a given mass, reducing the content of AM of the halo, the profile becomes steeper at the center. The steepening is easily explained in terms of an AM barrier. The halo’s central density is built up by shells having pericenters close to the halo center. All particles having large angular momenta are prevented from coming close to the halo’s center. As a consequence, they do not contribute to the central density, and this has the effect of flattening the density profile. Conversely, steeper central cusps are produced by more material having low AM.

This simple process has been discussed in several analytical papers (Nusser 2001; Hiotelis 2002; Le Delliou and Henriksen 2003; Ascasibar et al. 2004; Williams et al. 2004; DP09), and simulations (e.g., Polisenky and Ricotti 2015).

At the same time, higher peaks (larger $\nu$), which are progenitors of more massive haloes, have greater density contrast (deeper potential) at their center, and so the shells do not expand far before beginning to collapse. This reduces the AM, with respect to the standard SIM (Gunn and Gott 1972) and allows haloes to become more concentrated (see Sect. 3 of DP09, and Sect. 5 of Williams et al. 2004 for a wider discussion).

In summary, the shape of the halo density profile depends on its mass, and AM (also on baryon fraction, and dynamical friction). The relative role of mass and AM is shown in Fig. 1 of DP09; that of mass, AM, and baryon fraction is given in Figs. 1 and 2 of Del Popolo (2012a), and Fig. 1 of Del Popolo (2012b) (see also Fig. 1 of Williams et al. 2004), while that of mass, AM, and dynamical friction in Figs. 1 and 2 of Del Popolo et al. (2014).

As a consequence, a dwarf spiral, despite having less AM than a giant spiral, has a flatter profile because it has a shallower potential than a giant spiral. As discussed, in the formation phase, shells constituting the object can move far away from the center, acquire more AM, and as a consequence the DM particles tend to remain on larger orbits giving rise to flatter profiles. The same cannot happen in giant spirals having a deeper potential well.

This is similar to what happens in the supernovae feedback mechanism. In dwarf galaxies, the profile is flatter with respect to giant galaxies, even if the last have a larger content of stars. This is because in dwarf galaxies the expanding process of DM produced by SN explosions is not stopped or slowed down by the deep potential wells typical of giant galaxies.

The previous arguments explain why disc galaxies have shallower inner profiles than spheroids. In fact, for a given mass $M$, the inner slope depends on the AM of the galaxy: the larger it is, the flatter is the profile. Since spheroids have on average less AM than discs, this explains the difference in the inner density profile of the two classes of objects.

In Eq. (16), dynamical friction appears, which is calculated as described in Appendix D of DP09, and Antonuccio-Delogu and Colafrancesco (1994). As shown in DP09, the dynamical role of dynamical friction is similar to that of AM: the larger it is, the farther DM particles remain from the center of the structure. In the case of a peak with $\nu = 3$, Fig. 11 in DP09 shows that the effect of DF increases the collapse time of $\simeq 5~\%$ with respect to the standard SIM (Gunn and Gott 1972), and the joint effect of DF and AM produces an increase of $\simeq 15~\%$. Moreover, dynamical friction transfer energy of incoming clumps to DM.

When baryons cool dissipatively and collapse to the proto-structure center, the DM is compressed, with the result of a steepening of the DM density profile (Blumenthal et al. 1986; Gnedin et al. 2004; Gustafsson et al. 2006). The process is dubbed adiabatic contraction (AC) and counteracts the flattening produced by AM and dynamical friction.

In the present paper, the model used to evaluate AC is that of Blumenthal, improved following Gnedin et al. (2004), who showed that numerical simulation results are better reproduced if one assumes the conservation of the product of the mass inside the orbit-averaged radius, and the radius itself. The equations were solved through iterative techniques Spedicato et al. (2003).

As usual, it is assumed that the initial density profile of DM and baryons is the same (Mo et al. 1998; Keeton 2001; Treu and Koopmans 2002; Cardone and Sereno 2005) (i.e., an NFW profile), and the final distribution of baryons is assumed to be a disc (for spiral galaxies) (Blumenthal et al. 1986; Flores et al. 1993; Mo et al. 1998; Klypin et al. 2002; Cardone and Sereno 2005), the model by Klypin et al. (2002) (see their Sect. 2.1) for large spirals, and Hernquist model (Rix et al. 1997; Keeton 2001; Treu and Koopmans 2002) (see Appendix E of DP09 for details), for spheroidal and elliptical galaxies.

1.2 A.2 Baryons, discs, and clumps

Our model contains DM and baryons (initially) in the gas phase. The initial baryon fraction is set equal to the “universal baryon fraction” $f_{\mathrm{ub}}=0.17\pm 0.01$ (Komatsu et al. 2009) (0.167 in Komatsu et al. 2011).²¹

In the case of spiral galaxies, the infalling gas will settle into a disc, which is initially stable.

Concerning the disc sizes and masses obtained in our model, we have to recall that we have already dealt with this issue in Sect. 3.2 of Del Popolo et al. (2014), when we discussed how the angular momentum catastrophe (AMC) was solved in our model (see also Figs. 3 and 4 in the quoted paper). This produces discs (of size and mass) similar to those of real galaxies.

When the disc density increases, it is prone to instabilities. As is well known, stationary gas is unstable if the Jeans criterion is satisfied, while in the case of a rotating disc, the shear force can provide an additional stabilizing force. The condition for the disc instability, and that for the formation of an in-situ clump, is

$$ Q \simeq \sigma \Omega/(\pi G \Sigma)=\frac{c_{s} \kappa}{\pi G \Sigma}< 1, $$

(17)

see Toomre (1964), with $\sigma$ being the 1-D velocity dispersion,²² $\Omega$ the angular velocity, $\Sigma$ the surface density, $c_{s}$ the adiabatic sound speed, and $\kappa$ the epicyclic frequency. When $Q<1$, the fastest growing mode is obtained knowing the dispersion relation for a perturbation (see Binney and Tremaine 1987, or Nipoti and Binney 2015, Eq. (6)) from the solution of $d \omega^{2}/d k=0$, giving $k_{\mathrm{inst}}=\frac{\pi G \Sigma}{c_{s}^{2}}$.

The previous condition was applied to the discs of our galaxies. The clumps radii obtained are given by

$$ R \simeq 7 G \Sigma/\Omega^{2} \simeq 1~\mbox{kpc} $$

(18)

(see Krumholz and Dekel 2010).

For marginally unstable discs ($Q \simeq 1$), for which the velocity dispersion is maximal and the total mass is three times larger than the disc cold fraction, the clumps formed have a mass $\simeq 10~\%$ $M_{d}$ (Dekel et al. 2009).

Haloes of $5 \times 10^{11} M_{\odot}$ at $z \simeq 2$ have discs of few $\times 10^{10} M_{\odot}$, which remain for $\simeq 1~\mbox{Gyr}$ in a marginally unstable state, forming clumps, of masses $\simeq 10^{9} M_{\odot}$.

In the case of dwarf galaxies, clumps have masses of $\simeq 10^{5} M_{\odot}$.

The clumps characteristics (density, rotation velocity) are similar to those found by Ceverino et al. (2012) (see their Figs. 15 and 16).

Energy and AM transfer from clumps to DM can flatten the profile, and the process is the more efficient the earlier it happened, when haloes were smaller. The effectiveness of the process has been confirmed by several authors (Ma and Boylan-Kolchin 2004; Nipoti et al. 2004; Romano-Diaz et al. 2008, 2009; DP09; Cole et al. 2011; Inoue and Saitoh 2011; Del Popolo et al. 2014; Nipoti and Binney 2015).

1.2.1 A.2.1 Clumps life-time

An important issue is the clumps life-time. Clumps must live long enough to transform the cusp into a core.

Observations of galaxies at high redshift show the presence of clumpy structures, dubbed chain galaxies and clump clusters (e.g., Elmegreen et al. 2004, 2009; Genzel et al. 2011). Also HST Ultra Deep Field observations found massive star-forming clumps (Guo et al. 2012; Wuyts et al. 2013) present in a large number of star-forming galaxies at $z=1$–3 (Guo et al. 2015), and even in other deep fields up to $z \simeq 6$ (Elmegreen et al. 2007).

These clumpy structures should be formed from the instability of accreting gas (e.g., Noguchi 1998, 1999; Aumer et al. 2010; Ceverino et al. 2010, 2012), instability connected to the presence of a very gas-rich disc. Radiative cooling enhanced in the dense gas induces self-gravity instability and results in the clump formation.

According to the hydro-dynamical simulations of Ceverino et al. (2010), the clumps are in Jeans equilibrium and rotationally supported. This leads them to conclude that they are probably long-lived ($\simeq 2 \times 10^{8}~\mbox{Myr}$). Krumholz and Dekel (2010) showed that if the gas is converted in stars at a rate of a few percent, as in local systems forming stars, in agreement with the Kennicutt–Schmidt law, the clumps are not destroyed. The gas is retained and converted into stars remaining bound, and so the clump migrates to the galaxy center. Simulations by Elmegreen et al. (2008) reach the same conclusions of Ceverino et al. (2010) and Krumholz and Dekel (2010).

Last but not least, an important evidence in favor of the long lived clumps comes from observations of clumps at low redshift, having close similitude (radius, mass) to that observed in the high-redshift universe Elmegreen (2013), and Garland (2015) (see also Mandelker et al. 2015).

The previous results are very robust to changes in the lifetime, or in $\sigma$. Concerning the life-time, changes till an order of magnitude in the life-time, will not considerably change the results. As shown by Nipoti and Binney (2015), the flattening process happens on the dynamical friction time-scale, $1.4 t_{\mathrm{cross}}/\ln \Lambda$, where the crossing time for a system with $M_{\mathrm{gas}} \simeq 10^{7} M_{\odot}$, is $t_{\mathrm{cross}} \simeq 21~\mbox{Myr}$, and the Coulomb logarithm is $\simeq 2$, implying that the life-time is an order of magnitude larger than the flattening-time.

Concerning $\sigma$, as previously reported values in the range 20–80 (see footnote 23), satisfy the instability condition and clumps formation.

1.3 A.3 Star formation and feedback

Gas cooling, star formation, reionization, and supernovae feedback were included as done by De Lucia and Helmi (2008) and Li et al. (2010) (Sects. 2.2.2 and 2.2.3).

Reionization produces a reduction of the baryon content, whose fraction changes as

$$ f_{\mathrm{b}, \mathrm{halo}}(z,M_{\mathrm{vir}})=\frac{f_{\mathrm{b}}}{[1+0.26 M_{\mathrm{F}}(z)/M_{\mathrm{vir}}]^{3}} $$

(19)

(Li et al. 2010), where the reionization redshift is in the range 11.5–15, $M_{\mathrm{vir}}$ is the virial mass, and $M_{\mathrm{F}}$ is the “filtering mass” (see Kravtsov et al. 2004). Gas cooling is processed as a cooling flow (e.g., White and Frenk 1991) (see Sect. 2.2.2 of Li et al. 2010).

Concerning star formation, gas settles in a disk, and the star formation rate is

$$ \psi=0.03 M_{\mathrm{sf}}/t_{\mathrm{dyn}}, $$

(20)

which will give rise to an amount of stars

$$ \Delta M_{\ast}=\psi\Delta t, $$

(21)

where $M_{\mathrm{sf}}$ is the gas mass above a given density threshold, which is fixed as in DC14 as $n>9.3/{\mbox{cm}^{3}}$. $\Delta t$ indicates the time-step, and $t_{\mathrm{dyn}}$ is the disc dynamical time (see De Lucia and Helmi 2008 for more details).

Supernovae feedback is obtained as in Croton et al. (2006). When an SN explodes, it injects in the ISM a quantity of energy given by

$$ \Delta E_{\mathrm{SN}}=0.5\epsilon_{\mathrm{halo}}\Delta M_{\ast} \eta_{\mathrm{SN}}E_{\mathrm{SN}}, $$

(22)

where the number of supernovae per solar mass is $\eta_{\mathrm{SN}}=8\times 10^{-3}/M_{\odot}$, in the case of Chabrier IMF (Chabrier 2003), and $E_{\mathrm{SN}}=10^{51}$ erg is the typical energy released in an SN explosion.

After energy injection, the gas reheating is proportional to the stars formed

$$ \Delta M_{\mathrm{reheat}} = 3.5 \Delta M_{\ast}. $$

(23)

A quantity of hot gas equal to

$$ \Delta M_{\mathrm{eject}}=\frac{\Delta E_{\mathrm{SN}}-\Delta E_{\mathrm{hot}}}{0.5 V^{2}_{\mathrm{vir}}} $$

(24)

is ejected by the halo, if $\Delta E_{\mathrm{SN}}>\Delta E_{\mathrm{hot}}$, where

$$ \Delta E_{\mathrm{hot}}= 0.5\Delta M_{\mathrm{reheat}} \eta_{\mathrm{SN}}E_{\mathrm{SN}} $$

(25)

is the thermal energy change produced by the reheated gas.

We should stress that a fundamental difference between the DC14 SNFF model and our model is that the cusp flattening starts before star formation, and the energy source is gravitational. Stellar and supernovae feedback start when the core is in place, and those feedback processes act to disrupt the gas clouds that formed the core (similarly to Nipoti and Binney 2015). In contrast, the energy source in the SNFF processes is supernovae feedback, and the flattening process starts after stars form and explode.

Before concluding, I want to recall that we showed the robustness of the previous model in several ways:

• (a) In DP12a, the results were compared and found to be in agreement with the density profile of galaxies of $M \simeq 10^{10} M_{\odot}$ obtained in Governato et al. (2010) SPH simulations, and in Del Popolo and Hiotelis (2014) with the SPH simulations of Inoue and Saitoh (2011).

• (b) The results of the model are in agreement with several other studies of the DFBC mechanism (El-Zant et al. 2001, 2004; Romano-Diaz et al. 2008; Cole et al. 2011; Inoue and Saitoh 2011; Nipoti and Binney 2015).

• (c) In DP12b, the model predicted correlations and shapes of the density profiles of clusters then found by Newman et al. (2013a, 2013b). In DP10 and DP12b, it was shown that the inner slope depends on mass, in agreement with SPH simulations (e.g., DC14).

1.4 A.4 Dynamics of the satellites

As previously discussed, the flattening of galaxies is studied through the model described in Sect. A.1 (based on DP09, Del Popolo and Hiotelis 2014). In this model galaxies are interacting with neighbors only through tidal interaction.

In order to take into account the environmental effects to which satellites are subject during the infall in the host halo, together with the cusp–core transformation obtained with the model described in the previous sections, we follow Del Popolo and Le Delliou (2014). In the quoted paper, some of the small scale problems of the $\Lambda\mbox{CDM}$ model were studied combining the model of cusp–core transformation, described in Sect. A.1 (based on DP09 and Del Popolo and Hiotelis 2014) with a semi-analytic model that follows the substructure evolution within DM haloes. It takes into account the effects of DF, tidal stripping, and tidal heating on the infalling satellites. The model is basically the Taylor and Babul (2001) (TB01) model with small changes coming from a similar model by Peñarrubia et al. (2010) (P10).

The model is described in Del Popolo and Le Delliou (2014). Pullen et al. (2014) used a very similar model to study the evolution of dark matter only in subhaloes. That model is part of the code GALACTICUS (Benson 2012), a semi-analytic code of galaxy formation.

1.5 A.5 Formation of spiral and spheroids

Until recently, in the traditional picture, astrophysicists thought that the major morphological features of a galaxy are determined by the assembly history and net spin of its surrounding dark matter halo: whereas discs would form within quiescently built haloes whose AM content exceeds the average of the whole population, mergers would create early type elliptical galaxies.

Sales et al. (2012) cosmological hydro-dynamical simulations reached different conclusions. According to the quoted authors, discs form when the infalling gas flows in with similar AM to that of the material that accreted earlier, while spheroids form when the spin of the newly-accreted gas is misaligned with that of the galaxy. Galactic morphology is due to the interplay between the tidal field and the material which gives rise to the galaxy.

In our model, spheroids form as in the primordial monolithic collapse from the transformation of individual gas clouds of non-rotating gas into stars (top-down model) (Eggen et al. 1962 (ELS); Gunn and Gott 1972; Arimoto and Yoshii 1987) and variations thereof (Peacock et al. 1998).

Discs originate as in the monolithic collapse from the dissipational collapse of clouds with scant star formation (Larson 1976) endowed with high AM which is conserved in the collapse phase (Fall and Efstathiou 1980; Mo et al. 1998). As discussed in Sect. A.1, the final baryon configuration is fixed in the adiabatic contraction phase to be a disk or a spheroid. The previous basic model was improved following Sales et al. (2015) results. As in Sales et al. (2012), we measure morphology in terms of

$$ \kappa_{\mathrm{rot}} =\frac{K_{\mathrm{rot}}}{K}=\frac{1}{K} \sum \frac{m}{2} \biggl( \frac{j_{z}}{R} \biggr)^{2} $$

(26)

where $K_{\mathrm{rot}}$ is the kinetic energy in ordered motions, $j_{z}$ the component of AM orthogonal to the disk, $m$ the mass of a fraction of the system (e.g., the mass contained in a mass shell), and $R$ its radius. Discs with perfectly circular motions have $\kappa_{\mathrm{rot}} \simeq 1$, and spheroids have $\kappa_{\mathrm{rot}}\ll1$. As in Sales et al. (2012), disk dominated galaxies have $\kappa_{\mathrm{rot}}>0.7$, and spheroids $\kappa_{\mathrm{rot}}<0.5$. We then define, at turn-around, $\cos{\theta}$ as the angle, between the AM enclosed by different mass shells, $m$, and the total one, $m_{\mathrm{tot}}$ (see Sales et al. 2012). At turn-around, we calculate the ratio $m/m_{\mathrm{tot}}$, and $\cos{(\theta)}$ and we impose that $\cos{(\theta)}$ in terms of $m/m_{\mathrm{tot}}$ reproduces the behavior of the $\cos{(\theta)}$–$m/m_{\mathrm{tot}}$ relation given in Fig. 9 of Sales et al. (2012). We then choose, in the quoted systems, those that for an $M_{*}$ have $j_{*}$ in agreement with the $j_{*}$–$M_{*}$ relation of Fig. 1.

Concerning satellite dwarf galaxies, the action of tidal stripping, tidal heating, and ram-pressure stripping on a rotating late-type dwarf can remove the majority of its AM and gas transforming it in a non-rotating and quiescient early-type galaxy (Mayer et al. 2006; Schroyen et al. 2011). If tidal effects are not strong, and in absence of ram-pressure stripping, a late-type dwarf can be transformed into a dIrr/dE transition type dwarf (see the Introduction of Schroyen et al. 2011).

1.6 A.6 TF, FJ, and the $M_{*}$–$M_{\mathrm{halo}}$ relations

In this section, we compare the stellar mass–TF, FJ relation obtained by Dutton et al. (2010), and the $M_{*}$–$M_{\mathrm{halo}}$ relation of Moster et al. (2013), and Munshi et al. (2013) with the same relations obtained with the model of the present paper. In the top left panel of Fig. 4, the short dashed line represents the Dutton et al. (2010) TF, and the thin solid lines represents the uncertainty (0.027 on the zero point, 0.003 on the slope, and intrinsic scatter 0.05 dex on $V_{\mathrm{rot}}$).

Fig. 4

Stellar mass-TF, FJ relation, and $M_{*}$-$M_{\mathrm{halo}}$ relation. Left top panel: the short dashed line is the Dutton et al. (2010) TF, and the thin solid lines represents the uncertainty (0.05 dex in $V_{\mathrm{rot}}$). The thick solid line is the TF relation obtained using this paper model. The uncertainty is 0.071 dex in $\sigma$. Right top panel: as in the left panel but for the FJ relation. Bottom left panel: comparison of the baryonic TF relation of this paper (solid line), with that of McGaugh and Wolf (2010). The symbols are as follows: green squares (ultrafaint dwarfs (UFD)), dark gray points (star-dominated spirals), gas-dominated disks (light gray points), large blue squares (classical dwarfs), red triangles (M31 dwarfs), and the light blue star (Leo T). The dashed line is the best fit excluding the UFD (see McGaugh and Wolf 2010). Bottom right panel: the FJ relation of C04. Circles are observed galaxies, the other symbols are simulated galaxies. The dashed line connects the isolated galaxies, and the solid line is our FJ relation

The thick solid line represents the TFR obtained using the model used in the present paper. The plot shows good agreement of our TFR with that of Dutton et al. (2010).

The top right panel represents the comparison of our FJR with that of Dutton et al. (2010). Symbols are like the previous plot, and again good agreement is found. In this case, the uncertainty on the slope is 0.020, and the scatter is 0.071 dex in $\sigma(R_{50})$.

In order to make comparisons with smaller values of the mass, in the bottom left panel, we compare our TF relation with that of McGaugh and Wolf (2010), while in the bottom right panel we compare our FJ relation to that of C014 (see Fig. 4 caption for details).

In Fig. 5, we plot the stellar mass vs. halo mass relation and compare with those of Moster et al. (2013), and Munshi et al. (2013) (see Fig. 5 caption for details).

Fig. 5

Stellar mass vs. halo mass (SMH) relation. Dashed-line (Moster et al. 2013), solid line (our SMH), open dots (SMH with stellar masses measured directly from Munshi et al. 2013 simulations), and solid dots (SMH with stellar masses measured through Petrosian magnitudes, Munshi et al. 2013)

Appendix B: Galaxies in and around the Local Group

Apart for the MW and Andromeda, in the McConnachie’s sample there are two other spiral galaxies, namely M33 and NGC 300. dSph galaxies are predominant, followed in number by dIrr galaxies, mixed morphologies dIrr/dSph, and dE/dSph, a few Irregulars (Irrs), and one cE, namely M32.

In this paper, as MC12, and Grebel et al. (2003), galaxies fainter than $M_{V} \simeq -18$ are considered dwarf galaxies. Then all the galaxies in the sample, except for the MW, the LMC, NGC 55, M33, M31, and NGC 300, are dwarves.

dIrrs are irregular and gas-rich galaxies with $\mu_{V} \lesssim 23~\mbox{mag/arcsec}^{2}$, $M_{\mathrm{HI}} \lesssim 10^{9} M_{\odot}$, and $M_{t} \lesssim 10^{10} M_{\odot}$ (Grebel 2001). Low mass dIrss do not show measurable rotation, and could evolve to become dSphs, while the more massive have often rigid body rotation, but not shear, like the differential rotation in the discs of spirals, and may go on forming stars along all their history (Hunter 1997; Grebel 2001).

dEs have spherical or elliptical shape, and are fainter than $M_{V}=-17~\mbox{mag}$, have $\mu_{V} \lesssim 21~\mbox{mag/arcsec}^{2}$, $M_{\mathrm{HI}} \lesssim 10^{8} M_{\odot}$, and $M_{t} \lesssim 10^{9} M_{\odot}$ (Grebel 2001).

dSphs are gas-poor, and have a low-surface-brightness, $\mu_{V} \gtrsim 22~\mbox{mag/arcsec}^{2}$, $M_{V} \gtrsim -14~\mbox{mag}$, $M_{\mathrm{HI}} \lesssim 10^{5} M_{\odot}$, and $M_{t} \lesssim 10^{7} M_{\odot}$ (Grebel 2001). They are then the least massive and faintest galaxies known.

dIrr/dSphs, are transition-types and have $M_{\mathrm{HI}} \lesssim 10^{6} M_{\odot}$, much smaller than dIrrs ($10^{7} \lesssim M_{\mathrm{HI}}\lesssim 10^{9} M_{\odot}$).

Classifying objects into objects having measurable rotation or not is not an easy task. In the plane V-band luminosity–mean metallicity of red giants (see Fig. 1 of Grebel et al. 2003, and Fig. 7 of Mateo 1998), massive dIrrs and dSphs are grouped into different areas of the plane. dIrr/dSphs are generally closely located to dSphs, and they differ from dSphs only for the gas content. This is the case of the DDO 210, Phoenix, Antlia, LGS 3, and KKR 25. Peg DIG which is a transition type galaxy, differently from the previous ones, is located in the dIrr’s locus. GR 8 and Leo A are faint dIrrs, and differently from massive dIrrs are close to the dSphs locus. It is often assumed that if the ratio between the rotational velocity and velocity dispersion, $V_{\mathrm{rot}}/\sigma$, is $\lesssim 1$, random motion should dominate the rotation (e.g., Lo et al. 1993), and usually dSphs fulfill this criterion. However, there are exceptions to it (e.g., Phoenix).

Because of the similitude among low mass dIrrs, dIrr/ dSphs and dSphs, one can conclude that the more massive dIrrs are characterized by rigid body rotation, while the low mass dIrrs, and dIrr/dSphs have low rotational support (Grebel 2001; Grebel et al. 2003). Objects like GR 8, Leo A, LGS 3, Sag DIG, and DDO 210 do not show systematic rotation in HI (Lo et al. 1993; Young and Lo 1997).²³

In the case of Phoenix, $V_{\mathrm{rot}}/\sigma=1.7$, but the HI cloud has a small velocity gradient. This can be due to ejections from Phoenix (St-Germain et al. 1999), or due to rotation. Peg DIG has $V_{\mathrm{rot}}/\sigma=1.7$. In the case of KKR 25 and Antlia, the $V_{\mathrm{rot}}/\sigma$ is not measured, but the measurement of HI rotation indicates that $V_{\mathrm{rot}} \lesssim 5~\mbox{km/s}$ (Grebel et al. 2003).

Summarizing the previous discussion, dIrr/dSphs have similar characteristics to dSphs, and very small rotation. Massive dIrrs are rotationally supported, while small mass dIrrs are not.

One could then empirically “define” the boundary among rotation and no rotation by dIrrs such as GR 8, Leo A, LGS 3, SagDIG, and DDO 210. If one adopts the dynamical masses compiled by MC12, this would imply that galaxies with dynamical masses below a few times $10^{7} M_{\odot}$ are no longer rotation-dominated, whereas galaxies with masses $> 10^{8} M_{\odot}$ do show measurable rotation (Grebel, private communication). For precision’s sake, we should report that the dSphs dominate in systems like M31 or the MW, while in general isolated dwarf galaxies in the local universe have different characteristic. However, we already discussed that the $j_{*}$–$M_{*}$ relation in the case of isolated dwarfs has similar characteristics to that obtained from the MC12 sample.

Following the previous criteria, we divided the MC12 sample into fast rotators (FR) and slow rotators (SR). This is shown in Table 1. The first column gives the name of the galaxy, the second the stellar mass, the third the dynamical mass, the fourth the specific stellar AM, and the fifth the morphological (Hubble) type. The galaxies are divided into two groups (slow rotators and fast rotators, respectively upper and lower part of the table).

Table 1 Parameters for McConnachie (2012) galaxies. The first column gives the name of the galaxy, the second the stellar mass, the third the dynamical mass, the fourth the specific stellar AM, and the fifth the morphological (Hubble) type. The values of $j_{*}$ were calculated similarly to the way RF12 calculated the SAM for their sample, as described in Sect. 2

Figure 1 plots the distribution FRs (histogram in red) and SRs (histogram in blue).

The maximum mass of non-rotating objects is $10^{7.8} M_{\odot}$.

This mass threshold between rotation and non-rotation dominated galaxies should not be considered as an abrupt transition between the two regimes, but as the mass under which the AM gradually decreases from the values of discs to that of spheroids, as shown in Fig. 2 of RF12.

Appendix C: Tidal torque theory, and the $j_{*}$–$M_{*}$ relation

From the tidal torque theory (TTT) (Hoyle 1949; Peebles 1969; Doroshkevich 1970; White 1984) we know that $j \propto M^{2/3}$. They considered a simple model of transformations of gas into stars taken into account by the parameter $f_{*}$ (fraction of the initial gas that transforms in stars), and an idealized model of the baryon AM transfer, parameterized through $f_{j}$.²⁴

Using the two quoted parameters, the $j$–$M$ relation for dark matter will become the $j_{*}$–$M_{*}$ for stars:

$$ j_{*}=2.92 \times 10^{4} f_{j} f_{*}^{-2/3} \lambda \biggl(\frac{M_{*}}{10^{11} M_{\odot}} \biggr)^{2/3}~\mbox{km/s}\,\mbox{kpc}, $$

(27)

where $\lambda$ is the spin parameter, with $\lambda=\langle\lambda\rangle=0.035$. The average value of $f_{j} f_{*}^{-2/3}$ ($\langle f_{j} f_{*}^{-2/3}\rangle$) can be obtained using a sample of galaxies. RF12 found $\langle f_{j} f_{*}^{-2/3}\rangle \simeq 1.9$ for Sb-Sm galaxies, and $\langle f_{j} f_{*}^{-2/3}\rangle \simeq 0.5$ for ellipticals. A more precise analysis must take into account a dependency on the efficiency of star formation. This is obtained using the $\langle f_{\star}\rangle(M_{\star})$ given by Dutton et al. (2010). So, for galaxies with $M_{*} > 10^{8.6} M_{\odot}$, RF12 got the following $\langle f_{\star}\rangle(M_{\star})$:

$$ \langle f_{\star}\rangle(M_{\star}) = \frac{f_{0} (M_{\star}/M_{0} )^{1/2}}{ [1+ (M_{\star}/M_{0} ) ]^{1/2}} $$

(28)

for spiral galaxies. The constant $M_{0}$ is given by $\log (M_{0}/ M_{\odot}) \simeq 10.8$, and $f_{0} \simeq 0.33$. In the case of elliptical galaxies,

$$ \langle f_{\star}\rangle (M_{\star}) = \frac{f_{0} (M_{\star}/M_{0} )^{0.15}}{ [1+ (M_{\star}/M_{0} )^{2} ]^{1/2}}, $$

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where $\log (M_{0}/M_{\odot})\simeq11.2$, $f_{0} \simeq 0.14$.

Concerning $f_{j}$, RF12 considered two scenarios. In the first one, $f_{j}$ is variable and depending on morphology, based on the fact that the different $j_{*}$–$M_{*}$ relations of galaxies and spirals could be connected to different retentions in the SAM. In the second scenario, $f_{j}$ is constant, based on the fact that a scatter in the spin parameter, $\lambda$, is expected. Then they obtained average values for $f_{j}$, which are similar in the two scenarios.

As previously reported, discs and ellipticals are distributed on two bands with an offset in $j_{*}$.

RF12 and FR13 conclusions are confirmed by Teklu et al. (2015) and Zavala et al. (2015). In particular, in the second paper, the authors showed that spheroids are fundamentally formed by stars formed before turn-around, while discs are constituted by stars forming after turnaround.