Barions and ΛCDM Model Problems

Abstract

We examine the inner profiles of the rotation curves of galaxies in the velocity range from dwarf galaxies to Milky-way like galaxies, having maximum of rotation, ${{V}_{{{\text{max}}}}}$, in the range (30–250) km/s, whose diversity is much larger than that predicted by the $\Lambda$CDM model. After showing that the scatter in the observed rotation curves is much larger that predicted by dark matter-only cosmologies, we show how taking into account baryons, through a semi-analytical code, allows to create a a variety of rotation curves in agreement with observation. Simultaneously, we show that the quoted discrepancy does not need for different form of dark matter as advocated by other authors. We finally show how our model can reobtain the rotation curve of remarkable outliers like IC 2574, a 8 kpc cored profile having a challenging, and extremely low rising rotation curve, and UGC 5721 a cusp-like rotation curve galaxy. We suggest treating baryonic physics properly before introducing new exotic features, albeit legitimate, in the standard cosmological model.

Appendices

APPENDIX A

1.1 6. THEORETICAL MODEL

This section recalls the model employed in this work. The spherical collapse models [24–26, 78–80], was very significant improved in, e.g., [21, 81] to include the effects of

• random angular momentum induced by random motion during the collapse phase of haloes, e.g., [25, 26],

• ordered angular momentum induced by tidal torques, e.g., [27, 44, 82], and was furthered to include the consequences of

• adiabatic contraction, e.g., [28, 29, 83, 84],

• dynamical friction between DM and baryonic gas and stellar clumps [21, 30–33, 35, 85–87],

• gas cooling, star formation, photoionization, supernova, and AGN feedback [36, 37, 56] and

• DE [88–90],

and was further refined in [22, 23, 91, 92]. This model produced results on

• the universality of density profiles [59, 63],

• specific features of density profiles in

—galaxies [60, 93] and

—clusters [54, 93]

as well as a focus on

• galaxies inner surface-density [94].

Although the model’s key mechanism resides in dynamical friction (DFBC), we stress out that it includes all of the above effects (including SNF) that each only contribute at the level of some %.

Its implementation occurs in several stages:

1. The diffuse proto-structure of gas and DM expands, in the linear phase, to a maximum radius before DM re-collapses into a potential well, where baryons will fall.

2. In their radiative clumping, baryons form stars at the halo centre.

3. Then four effects happen in parallel:

(a) The DM central cusp increases from baryons adiabatic contraction (at $z \simeq 5$ in the case of ${{10}^{9}}{\kern 1pt} {{M}_{ \odot }}$ galaxies [21]).

(b) The galactic centre also receive clumps that collapse from baryons-DM dynamical friction (DF).

(c) The DF energy and angular momentum (AM) transfer to DM and stars [95–97] results in an opposite effect to adiabatic contraction, and reduces the halo central density [30, 31].

(d) The balance between adiabatic contraction and DF can result in heating cusps and forming cores, i.e. in dwarf spheroidals and spirals, while the deeper potential wells of giant galaxies keeps their profile steeper.

4. The effect of DF adds to that of tidal torques (ordered AM), and random AM.

5. Finally, the core further slightly (few percent) enlarges from the decrease of stellar density due to successive gas expulsion from supernovae explosions, and from the disruption of the smallest gas clumps, once they have partially converted to stars, see [35].

1.1.1 6.1. Model Treatment of Density Profile

Starting from a Hubble expansion, the spherical model of density perturbations expands linearly until reaching a turn-around maximum and reverting into collapse [98, 99]. A Lagrange particle approach yields the final density profile

$$\rho (x) = \frac{{{{\rho }_{{{\text{ta}}}}}({{x}_{{\text{m}}}})}}{{f{{{({{x}_{{\text{i}}}})}}^{3}}}}{{\left[ {1 + \frac{{d\ln f({{x}_{{\text{i}}}})}}{{d\ln g({{x}_{{\text{i}}}})}}} \right]}^{{ - 1}}},$$

(1)

with initial and turn-around radius, resp. ${{x}_{{\text{i}}}}$ and ${{x}_{{\text{m}}}}({{x}_{{\text{i}}}})$, collapse factor $f({{x}_{{\text{i}}}}) = x{\text{/}}{{x}_{{\text{m}}}}({{x}_{{\text{i}}}})$, and turnaround density ${{\rho }_{{{\text{ta}}}}}({{x}_{{\text{m}}}})$. The turn-around radius is obtained with

$${{x}_{{\text{m}}}} = g({{x}_{{\text{i}}}}) = {{x}_{{\text{i}}}}\frac{{1 + {{{\bar {\delta }}}_{{\text{i}}}}}}{{{{{\bar {\delta }}}_{{\text{i}}}} - (\Omega _{{\text{i}}}^{{ - 1}} - 1)}},$$

(2)

where we used ${{\Omega }_{{\text{i}}}}$ for the density parameter, and ${{\bar {\delta }}_{{\text{i}}}}$ for the average overdensity inside a DM and baryons shell.

The model starts with all baryons in gas form with ${{f}_{{\text{b}}}} = 0.17 \pm 0.01$ for the “universal baryon fraction” [100], set to 0.167 in [4], before star formation proceeds as described below.

Tidal torque theory (TTT) allows to compute the “specific ordered angular momentum,” $h$, exerted on smaller scales from larger scales tidal torques [27, 40–42, 66, 101–104], while the “random angular momentum,” $j$, is related to orbits eccentricity $e = \left( {\frac{{{{r}_{{{\text{min}}}}}}}{{{{r}_{{{\text{max}}}}}}}} \right)$ [105], obtained from the apocentric radius ${{r}_{{{\text{max}}}}}$, the pericentric radius ${{r}_{{{\text{min}}}}}$ and corrected from the system’s dynamical state effects advocated by [80], using the spherically averaged turnaround radius ${{r}_{{{\text{ta}}}}} = {{x}_{{\text{m}}}}({{x}_{{\text{i}}}})$ and the maximum radius of the halo ${{r}_{{{\text{max}}}}} < 0.1{{r}_{{{\text{ta}}}}}$

$$e({{r}_{{{\text{max}}}}}) \simeq 0.8{{\left( {\frac{{{{r}_{{{\text{max}}}}}}}{{{{r}_{{{\text{ta}}}}}}}} \right)}^{{0.1}}}.$$

(3)

These corrections to the density profile are compounded also with its steepening from the adiabatic compression following [29] and the effect of DF introduced in the equation of motion by a DF force, see [21, Eq. A14].

1.1.2 6.2. Effects of Baryons, Discs, and Clumps

The baryon gas halo settles into a stable, rotationally supported, disk, in the case of spiral galaxies. Their size and mass result from solving the equation of motion, and lead to a solution of the angular momentum catastrophe (AMC) [93, Sec. 3.2, Figs. 3, and 4], obtaining realistic disc size and mass.

Notwithstanding stabilization from the shear force, Jean’s criterion shows the appearance of instability for denser discs. The condition for this appearance and subsequent clump formation was found by Toomre [106], involving the 1-D velocity dispersion $\sigma$,³ angular velocity $\Omega$, surface density $\Sigma$, related to the adiabatic sound speed ${{c}_{s}}$, and the epicyclic frequency $\kappa$

$$Q \simeq \sigma \Omega {\text{/}}(\pi G\Sigma ) = \frac{{{{c}_{s}}\kappa }}{{\pi G\Sigma }} < 1.$$

(4)

The solution to the perturbation dispersion relation $d{{\omega }^{2}}{\text{/}}dk = 0$ for $Q < 1$ yields the fastest growing mode ${{k}_{{{\text{inst}}}}} = \frac{{\pi G\Sigma }}{{c_{s}^{2}}}$ (see [107] or [35, Eq. 6]). That condition allows to compute the clumps radii in galaxies [108]

$$R \simeq 7G\Sigma {\text{/}}{{\Omega }^{2}} \simeq 1\;{\text{kpc}}.$$

(5)

Marginally unstable discs ($Q \simeq 1$) with maximal velocity dispersion have a total mass three times larger than that of the cold disc, and form clumps $\simeq {\kern 1pt} 10$% of their disk mass ${{M}_{d}}$ [109–114].

Objects of masses few times ${{10}^{{10}}}{\kern 1pt} {{M}_{ \odot }}$, found in $5 \times {{10}^{{11}}}\,{{M}_{ \odot }}$ haloes at $z \simeq 2$, are in a marginally unstable phase for $\simeq {\kern 1pt} 1$ Gyr. Generally the main properties of clumps are similar to those found by [47].

In agreement with [21, 32, 33, 35, 85–87, 115, 116], energy and AM transfer from clumps to DM flatten the profile more efficiently in smaller haloes.

6.2.1. Computing the clumps life-time. Evidence for existence of the clumps produced by the model can be traced both in simulations, e.g., [117–123], and observations. High redshift galaxies have been found to contain clump clusters or clumpy structures that leads to call them chain galaxies, e.g., [124–126]. The HST Ultra Deep Field encompasses galaxies with massive star-forming clumps [127, 128] many at $z = 1{-} 3$ [129], some in deeper fields $z \leqslant 6$ [130].

Such clumpy structures are expected to originate from self-gravity instability in very gas-rich disc, from radiative cooling in the acreting dense gas, e.g., [47, 117, 131–133]. Their effect on halo central density depend crucially on the clump lifetime: should their disruption through stellar feedback still allow them sufficient time to sink to the galaxy centre, they can turn a cusp into a core. A clump’s ability to form a bound stellar system is assessed through its stellar feedback mass fraction loss, ${{e}_{f}}$, and its formed stars mass fraction, $\varepsilon = 1 - {{e}_{f}}$. Simulations and analytical models agree that most of the mass of such group of stars will remain bound for $\epsilon \geqslant 0.5$ [134]. The radiation feedback efficiency can be estimated, using

(a) the dimensionless star-formation rate efficiency ${{\epsilon }_{{{\text{eff}}}}} = \frac{{{{{\dot {M}}}_{*}}}}{{M{\text{/}}{{t}_{{{\text{ff}}}}}}}$, which is simply the ratio between free-fall time, ${{t}_{{{\text{ff}}}}}$, and the depletion time for a stellar mass ${{M}_{*}}$. In its reduced version it reads ${{\epsilon }_{{{\text{eff}}}}}{{,}_{{ - 2}}} = {{\varepsilon }_{{{\text{eff}}}}}{\text{/}}0.01$;

(b) the reduced dimensionless surface density ${{\Sigma }_{1}} = \frac{\Sigma }{{0.1\;{\text{g/c}}{{{\text{m}}}^{2}}}}$,

and to obtain the expulsion fraction ${{e}_{f}} = 1 - \varepsilon =$ $0.086({{\Sigma }_{1}}{{M}_{9}}{{)}^{{ - 1/4}}}{{\epsilon }_{{{\text{eff}}}}}{{,}_{{ - 2}}}$ [108]. Authors in [135] estimated, for a large sample of environments, densities, size and scales, that ${{\epsilon }_{{{\text{eff}}}}} \simeq 0.01$. Furthermore, ${{e}_{f}} = 0.15$ and $\varepsilon = 0.85$ for typical clumps with masses $M \simeq {{10}^{9}}\,{{M}_{ \odot }}$. Therefore, the clump mass loss before they reach the centre of the galactic halo should be small. However, such conclusion and the expulsion fraction method are valid for smaller, more compact clumps in smaller galaxies. Such context only produces clumps that survive all the way to the centre.

Alternately, comparing a clump lifetime to its migration time to the centre, one can also obtain clump disruption. Migration time is the result of DF and TTT: for a ${{10}^{9}}{\kern 1pt} {{M}_{ \odot }}$ clump, it yields $\simeq {\kern 1pt} 200$ Myrs, see [35, Eq. (1)] and [126, Eq. (18)]. Coincident expansion and migration timescales were computed from the Sedov–Taylor solution [126, Eqs. (8), (9)].

Clump lifetime has been much studied. Ceverino et al., finding clumps in Jean’s equilibrium and rotational support, from hydrodynamical simulations [117], construed their long lifetime ($\simeq {\kern 1pt} 2 \times {{10}^{8}}$ Myr). This agrees with several approaches: in local systems forming stars and coinciding with the Kennicutt–Schmidt law, [108] found such lifetimes. This is because as clumps retained gas, and formed bound star groups, they had time to migrate to the galactic centre. Simulations from [136] confirmed it. Other simulations with proper account of stellar feedback, e.g., non-thermal and radiative feedback mechanisms, also obtained long-lived clumps reaching galactic centre (SNF, radiation pressure, etc. [119, 120, 122]). Finally, the same was obtained with any reasonable amount of feedback [118]. The expansion, gas expulsion, and metal enrichment, time scales (respectively, $> {\kern 1pt} 100$ Myrs, 170–1600 Myrs, and $\simeq {\kern 1pt} 200$ Myrs) obtained by [126] to estimate clump ages also bring strong evidence for long-lived clumps. Lastly, comparison between similar low and high redshift clumps observations, in radius, mass [137–139] supports clump stability.

1.1.3 6.3. Model Treatment of Feedback and Star Formation

Star formation, reionization, gas cooling, and SNF in the model are built along [36, 37, Sections 2.2.2 and 2.2.3].

Reionization acts for $z = 11.5{-} 15$ by decreasing the baryon fraction as

$${{f}_{{{\text{b,halo}}}}}(z,{{M}_{{{\text{vir}}}}}) = \frac{{{{f}_{{\text{b}}}}}}{{{{{[1 + 0.26{{M}_{{\text{F}}}}(z){\text{/}}{{M}_{{{\text{vir}}}}}]}}^{3}}}},$$

(6)

[37] using the virial mass,${{M}_{{{\text{vir}}}}}$, and the “filtering mass” [see 140], ${{M}_{{\text{F}}}}$.

Gas cooling follows from a cooling flow model, e.g., [37, 141, Section 2.2.2].

Star formation arises from gas conversion into stars when it has settled in a disk. The gas mass conversion into stars during a given time interval $\Delta t$, which we take as the disc dynamical time ${{t}_{{{\text{dyn}}}}}$, is given by

$$\Delta {{M}_{*}} = \psi \Delta t,$$

(7)

where the star formation rate $\psi$ comes from the gas mass above the density threshold $n > 9.3{\text{/c}}{{{\text{m}}}^{3}}$ [fixed as in 52] according to [36], for more details

$$\psi = 0.03{{M}_{{{\text{sf}}}}}{\text{/}}{{t}_{{{\text{dyn}}}}}.$$

(8)

SNF follows [54], where SN explosions inject energy in the system. This energy can be calculated from a Chabrier IMF [53], using

—the disc gas reheating energy efficiency ${{\epsilon }_{{{\text{halo}}}}}$,

—the available star mass $\Delta {{M}_{ * }}$,

—that mass conversion into SN measured with the SN number per solar mass as ${{\eta }_{{{\text{SN}}}}} = 8 \times {{10}^{{ - 3}}}{\text{/}}{{M}_{ \odot }}$, and

—the typical energy an SN explosion releases ${{E}_{{{\text{SN}}}}} = {{10}^{{51}}}$ erg,

to obtain

$$\Delta {{E}_{{{\text{SN}}}}} = 0.5{{\epsilon }_{{{\text{halo}}}}}\Delta {{M}_{*}}{{\eta }_{{{\text{SN}}}}}{{E}_{{{\text{SN}}}}}.$$

(9)

This released energy from SNs into the hot halo gas in the form of reheated disk gas then compares with the reheating energy $\Delta {{E}_{{{\text{hot}}}}}$ which that same amount of gas should acquire if its injection in the halo should keep its specific energy constant, that is if the new gas would remain at equilibrium with the halo hot gas. That amount of disk gas the SN and stars radiation have reheated, $\Delta {{M}_{{{\text{reheat}}}}}$, since it is produced from stars radiations, is proportional to their mass

$$\Delta {{M}_{{{\text{reheat}}}}} = 3.5\Delta {{M}_{*}}.$$

(10)

Since the halo hot gas specific energy corresponds to the Virial equilibrium specific kinetic energy $\frac{{V_{{{\text{vir}}}}^{2}}}{2}$, keeping this energy constant under addition of that reheated gas leads to define the equilibrium reheating energy as

$$\Delta {{E}_{{{\text{hot}}}}} = 0.5\Delta {{M}_{{{\text{reheat}}}}}V_{{{\text{vir}}}}^{2}.$$

(11)

The comparison with the actual energy of the gas injected from the disk into the halo by SNs gives the threashold ($\Delta {{E}_{{{\text{SN}}}}} > \Delta {{E}_{{{\text{hot}}}}}$) beyond which gas is expelled, the available energy to expel the reheated gas, and thus the amount of gas ejected from that extra energy

$$\Delta {{M}_{{{\text{eject}}}}} = \frac{{\Delta {{E}_{{{\text{SN}}}}} - \Delta {{E}_{{{\text{hot}}}}}}}{{0.5V_{{{\text{vir}}}}^{2}}}.$$

(12)

Contrary to SNF based models such as [52], our mechanism for cusp flattening initiates before the star formation epoch. Since it uses a gravitational energy source, it is thus less limited in available time and energy. Only after DF shapes the core can Stellar and SN feedback occurs, which then disrupt gas clouds in the core similarly to [35].

AGN feedback occurs when a central Super-Massive-Black-Hole (SMBH) is formed. We follow the prescriptions of [56, 57], modifying the [58] model for SMBH mass accretion and AGN feedback: a seed ${{10}^{5}} {{M}_{ \odot }}$ SMBH forms when stellar density, reduced gas density (${{\rho }_{{{\text{gas}}}}}{\text{/}}10$) and 3D velocity dispersion exceed the thresholds $2.4 \times {{10}^{6}}\,{{M}_{ \odot }}{\text{/kp}}{{{\text{c}}}^{3}}$ and 100 km/s, which then accretes. Significant AGN quenching starts above $M \simeq 6 \times {{10}^{{11}}}{\kern 1pt} {{M}_{ \odot }}$ [55].

1.1.4 6.4. Model Robustness

We point out that the model demonstrated its robustness in various behaviors:

(a) The cusp flattening from DM heating by collapsing baryonic clumps predicted for galaxies and clusters is in agreement with following studies [30–33, 35, 86, 87]. A comparison with [61]’s SPH simulations was made in [63, Fig. 4].

(b) It aforetime predicted the correct shape of galaxies density profiles [21, 81] ahead of SPH simulations of [61, 62], and of clusters density profiles [64] anteriorly of [65].⁴

(c) It aforetime predicted the halo mass dependence of cusps inner slope [59, Fig. 2a solid line] beforehand the similar result in the non-extrapolated part of the plot in [52, Fig. 6], expressed in terms of ${{V}_{c}}$, as it corresponds to $2.8 \times {{10}^{{ - 2}}}M_{{{\text{vir}}}}^{{0.316}}$ [142].

(d) It also preceded [52] in predicting [64] that the inner slope depends on the total baryonic content to total mass ratio.

(e) It compares well with simulations [52] on the inner slope change with mass [22, 23, Fig. 1].

(f) It moreover provides a comparison of the Tully–Fisher and Faber–Jackson, ${{M}_{{{\text{Star}}}}}{-} {{M}_{{{\text{halo}}}}}$, relationships with simulations [22, 23, Figs. 4, 5].

APPENDIX B

1.1 7. DENSITY PROFILE DEPENDENCE FROM DIFFERENT PARAMETERS

One point that we want to discuss here is to show how the density profile, that is related to the rotation curve by the relation

$$\rho (R) = \frac{1}{{4\pi G}}\left[ {2\frac{V}{R}\frac{{dV}}{{dR}} + {{{\left( {\frac{V}{R}} \right)}}^{2}}} \right],$$

(13)

[143] changes according to different quantities (parameters). In Fig. 4, we show the effect of tidal angular momentum on the density profile. Figures 4a–4c correspond to masses ${{10}^{8}}\,{{M}_{ \odot }}$, ${{10}^{9}}\,{{M}_{ \odot }}$, and ${{10}^{{10}}}\,{{M}_{ \odot }}$, respectively. The angular momentum, h, in Fig. 4a is equal to that used in [21], ${{h}_{*}}$, that of Fig. 4b is $2{{h}_{*}}$, with the effect of flattening the profile, and that in Fig. 4c is ${{h}_{*}}{\text{/}}2$ with the effect of steepening the profile. Panel (d) shows the density profile of a halo of ${{10}^{{10}}}\,{{M}_{ \odot }}$ with zero ordered angular momentum and no baryons (solid line), while the dashed line is the Einasto profile. In Figs. 4a–4c the baryon fraction is ${{f}_{{{{d}_{ * }}}}} \simeq 0.04$, while as already told in Fig. 4d it is zero. The smaller is the baryon fraction the flatter is the profile as also shown in [64]. Concerning the role of the random angular momentum in shaping the profile, it is similar to that of the tidal angular momentum. In fact as shown in Fig. 9 of DP09 [21] the tidal torque angular momentum and the random one are directly proportional. As can be deduced from the quoted plot, the random angular momentum has a stronger effect that the tidal angular momentum, but a similar effect: larger random momentum flatten the profile, and smaller steepens it. Because of the similitude of effect of the two kind of angular momentum on the profile, we do not plot the effect of random angular momentum. Figure 5 shows how changing the baryon fraction modifies the profile.

Fig. 4.

Shape changes of dark matter haloes with angular momentum. In panels (a)–(c), the dashed line, the dotted line, and the solid line represent the density profile for a halo of ${{10}^{8}}{\kern 1pt} {{M}_{ \odot }}$, ${{10}^{9}}{\kern 1pt} {{M}_{ \odot }}$, and ${{10}^{{10}}}{\kern 1pt} {{M}_{ \odot }}$, respectively. In case (a), that is our reference case, the specific angular momentum was obtained using the tidal torque theory as described in Del Popolo (2009) (DP09) [21]. The specific angular momentum for the halo of mass ${{10}^{9}}{\kern 1pt} {{M}_{ \odot }}$ is ${{h}_{*}} \simeq 400$ kpc km/s ($\lambda \simeq 0.04$) and the baryon fraction ${{f}_{{{{d}_{ * }}}}} \simeq 0.04$. In panel (b) we increased the value of specific ordered angular momentum, ${{h}_{ * }}$, to $2{{h}_{ * }}$ leaving unmodified the baryon fraction to ${{f}_{{{{d}_{ * }}}}}$ and in panel (c) the specific ordered angular momentum is ${{h}_{ * }}{\text{/}}2$ and the baryon fraction equal to the previous cases, namely, ${{f}_{{{{d}_{ * }}}}}$. Panel (d) shows the density profile of a halo of ${{10}^{{10}}}{\kern 1pt} {{M}_{ \odot }}$ with zero ordered angular momentum and no baryons (solid line), while the dashed line is the Einasto profile.

Fig. 5.

DM halos shape changes with baryon fraction. Same as previous figure, Fig. 4, but in panel (a) we reduced the value of baryon fraction of Fig. 4a (${{h}_{ * }}$, ${{f}_{{{{d}_{ * }}}}}$) to ${{f}_{{{{d}_{ * }}}}}{\text{/}}3$, and in panel (b) we reduced the value of baryon fraction of Fig. 4c to ${{f}_{{{{d}_{ * }}}}}{\text{/}}3$; in panel (c) we increased the value of baryon fraction of Fig. 4a (${{h}_{ * }}{\text{/}}2$, ${{f}_{{{{d}_{ * }}}}}$) to $3{{f}_{{{{d}_{ * }}}}}$, and in panel (d) we increased the value of baryon fraction of Fig. 1c to $3{{f}_{{{{d}_{ * }}}}}$.

DM halos shape changes with baryon fraction. In Fig. 5a the value of the tidal angular momentum is the same as Fig. 4a but the value of baryon fraction of Fig. 4a (${{f}_{{{{d}_{ * }}}}}$) is reduced to ${{f}_{{{{d}_{ * }}}}}{\text{/}}3$, and in Fig. 5b we reduced the value of baryon fraction of Fig. 1c, that was ${{f}_{{{{d}_{ * }}}}}{\text{/}}$, to ${{f}_{{{{d}_{ * }}}}}{\text{/}}3$; in panel (c) we increased the value of baryon fraction of Fig. 4a (${{h}_{*}}{\text{/}}2$, ${{f}_{{{{d}_{ * }}}}}$) to $3{{f}_{{{{d}_{ * }}}}}$, and in panel (d) we increased the value of baryon fraction of Fig. 4c to $3{{f}_{{{{d}_{ * }}}}}$. Summarizing, increasing the baryon fraction the profile flattens and viceversa. Figure 6 shows the effect of dynamical friction. Figure 6a corresponds to Fig. 4c (${{h}_{*}}{\text{/}}2$) but having increased the coefficient of dynamical friction to the double of Fig. 4c. In Fig. 6b, the coefficient of dynamical friction is reduced to one half. Dynamical friction has a similar effect of that of angular momentum, as also shown in Fig. 1 of DP09 [21].

Fig. 6.

Dependence of the density profiles shape on dynamical friction. Figure 6a corresponds to Fig. 4c (${{h}_{*}}{\text{/}}2$) but having increased the coefficient of dynamical friction to the double of Fig. 4c. In Fig. 6b, the coefficient of dynamical friction is reduced to one half.