High-Redshift Star-Forming Galaxies: Angular Momentum and Baryon Fraction, Turbulent Pressure Effects and the Origin of Turbulence
A. Burkert, R. Genzel, N. Bouche, G. Cresci, S. Khochfar, J. Sommer-Larsen, A. Sternberg, T. Naab, N. Foerster-Schreiber, L. Tacconi, K. Shapiro, E. Hicks, D. Lutz, R. Davies, P. Buschkamp, S. Genel
aa r X i v : . [ a s t r o - ph . C O ] N ov High-Redshift Star-Forming Galaxies: Angular Momentum and BaryonFraction, Turbulent Pressure Effects and the Origin of Turbulence
A. Burkert , , R. Genzel , N. Bouch´e , G. Cresci , S. Khochfar , J. Sommer-Larsen , , A.Sternberg , T. Naab , N. F¨orster Schreiber , L. Tacconi , K. Shapiro , E. Hicks , D. Lutz , R.Davies , P. Buschkamp , S. Genel [email protected], [email protected] ABSTRACT
The structure of a sample of high-redshift ( z ∼ σ ≈ −
80 km/s is investigated. Fit-ting the observed disk rotational velocities and radii with a Mo, Mao & White (1998)(MMW) model requires unusually large disk spin parameters λ d > . d ≈ .
2, close to the cosmic baryon fraction. The galaxiessegregate into dispersion-dominated systems with 1 ≤ v max /σ ≤
3, maximum rota-tional velocities v max ≤
200 km/s and disk half-light radii r / ≈ v max >
200 km/s, v max /σ > r / ≈ λ d = 0 . − .
05 for high disk mass fractions of m d ≈ . λ d = 0 . − .
03 form d ≈ .
05. These values are in good agreement with cosmological expectations. Forthe rotation-dominated sample however pressure effects are small and better agreementwith theoretically expected disk spin parameters can only be achieved if the dark halomass contribution in the visible disk regime (2 − × r / ) is smaller than predicted by theMMW model. We argue that these galaxies can still be embedded in standard cold darkmatter halos if the halos did not contract adiabatically in response to disk formation.In this case, the data favors models with small disk mass fractions of m d = 0 .
05 and University Observatory Munich (USM), Scheinerstrasse 1, 81679 Munich, Germany Max-Planck-Fellow Max-Planck-Institut f¨ur extraterrestrische Physik (MPE), Giessenbachstr. 1, 85748 Garching, Germany Dark Cosmology Centre, Niels Bohr Institute, University of Copenhagen, Juliane Marie Vej 30, 2100 Copenhagen,Denmark Excellence Cluster Universe, Technical University Munich, Boltzmannstr. 2, D-85748 Garching, Germany School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel Department of Astronomy, Campbell Hall, University of California, Berkeley, CA 94720, USA λ d ≈ . Subject headings: cosmology: observations – galaxies: high-redshift – galaxies: individ-ual (BzK-15504) – galaxies: formation – galaxies: evolution – galaxies: halos
1. Introduction
Deep surveys have become efficient in detecting star-forming galaxy populations at z ∼ ⊙ /yr, with a range of ages (10 Myrs - 3 Gyrs), stellarmasses of M ∗ ∼ − . M ⊙ (Shapley et al. 2005; F¨orster Schreiber et al. 2006; Erb et al.2006a,b; Daddi et al. 2004a,b) and high gas fractions (Tacconi et al. 2010). They contribute alarge fraction of the cosmic star formation activity and stellar mass density at z ∼ ∼ α line emission has shown thatmost of these high-z star forming galaxies are clumpy and exhibit large ionized gas velocity dis-persions of 30-120 km/s (F¨orster Schreiber et al. 2006, 2009; Genzel et al. 2006, 2008; Law et al.2007, 2009; Wright et al. 2007, 2009; Van Starkenburg et al. 2008; Stark et al. 2008; Bournaud et al.2008; Epinat et al. 2009; Cresci et al. 2009). About one third appear to be rotating disks, onethird are dispersion dominated systems and one third show clear evidence for interactions and majormergers (Shapiro et al. 2008; F¨orster Schreiber et al. 2009). The fraction of large, clumpy rotat-ing disks increases with mass. The ratio of the rotational to random velocities ranges between 1 and6, quite in contrast to z ∼ σ ∼ ∼ σ ∼
20. The situation is however different at high redshifts where turbulence can stronglyaffect the disk structure. This paper discusses the impact of large turbulent motions on the inter-pretation of the dynamical data of disk galaxies. We show that, including turbulent pressure, thedisk spin parameters and disk mass fractions of dispersion-dominated galaxies are reduced to valuesthat are consistent with theoretical expectations. The situation is different for rotation-dominatedgalaxies where pressure effects play a minor role. As already suggested by numerous studies at lowredshifts (e.g. Mo & Mao (2000); Dutton et al. (2007)), we argue that the observed high-redshiftgalaxies are in better accord with cosmological models if it is assumed that their dark-matter halosdid not contract adiabatically. Finally we propose an explanation for why large turbulence mightbe more common in many high-z disks and what the energy source of turbulence in these disksmight be.
2. Rotation Curves of Pressurized, Turbulent Galactic Disks
Let us consider a turbulent galactic gas disk. We analyse its rotational velocity v rot in themidplane, applying the hydrostatic equation v rot r = f g ( r ) + 1 ρ dpdr (1)where r is the distance from the galactic center and f g is the value of the gravitational force. p isthe pressure which consists of a turbulent (kinetic) and thermal part, p = ρ ( σ + c s ) with ρ thegas density, σ the characteristic 1-dimensional velocity dispersion of the gas which we assume to beisotropic and c s its sound speed. We define the zero-pressure velocity curve v ( r ) as the rotationalvelocity of the gas if pressure gradients are negligible, i.e. dp/dr=0: v ≡ f g × r . Equation 1 thenreduces to v rot = v + rρ dpdr = v + 1 ρ ddlnr (cid:0) ρσ (cid:1) . (2) 4 –Here we have neglected the thermal pressure term as the sound speed is in general much smallerthan the turbulent velocity. Equation 2 is the most general form, without specifying the radialdependence of σ and ρ . It demonstrates that a negative radial pressure gradient reduces therotational velocity of the gas as part of the gravitational force is balanced by the pressure force.To illustrate the possible importance of pressure effects, let us now assume that σ is independentof r . Then v rot = v + σ dlnρdlnr . (3)If σ is also independent of height z above the disk’s equatorial plane, the vertical density distributionis given by the vertical hydrostatic Spitzer solution (Spitzer 1942; Binney & Tremaine 08, pagechapter 4, p. 390) ρ ( z ) = ρ sech ( z/h ) (4)with ρ (r) the density in the midplane (z=0) at radius r and h = σ √ πGρ (5)the scale height. The total mass surface density Σ( r ) of such a disk is (Binney & Tremaine 2008)Σ = 2 ρ h (6)so that ρ = πG Σ σ (7)The equations 5-7 offer an interesting future observational test of our assumptions as for an ex-ponential disk (equation 10) with constant velocity dispersion the scale height h should increaseexponentially with radius h = σ πG Σ exp (cid:18) rr d (cid:19) (8)Inserting ρ from equation 7 into equation 3, the rotation curve in the equatorial plane of a pres-surized gas disk is v rot = v + 2 σ dln Σ dlnr (9) 5 –For example, for an exponential disk profile with scale length r d Σ( r ) = Σ × exp (cid:18) − rr d (cid:19) (10)Equation 9 leads to v rot = v − σ (cid:18) rr d (cid:19) (11)Note that v rot is the actually observable rotational velocity of the gas, while v is the rotationexpected if pressure effects are negligible ( σ = 0). For v rot /σ . r & r d . A very similar equation holds for stellardisks as a special form of the Jeans equation (Binney & Tremaine 2008). Equation 11 was derivedfor an exponential surface density distribution and a constant velocity dispersion. In general, thesituation is more complex as parts of the disk might have constant surface densities while otherparts might show a steep gradient. In addition, the velocity dispersion might change with radius.In this case one would have to solve equation (1) directly. The best-resolved high-redshift diskgalaxies show roughly constant velocity dispersion profiles (Genzel et al. 2008) and exponentiallydecreasing H α -surface brightness distributions of the star-forming gas with scale lengths similar tothe stellar disks within 1-3 disk scale lengths (Cresci et al. 2009; Bouch´e et al., in preparation).This is also the region where the rotation curves are flat and achieve their maximum values v max (Fig. 3). v max will be used in the following to compare the observations with theory.For the purpose of this analyses we adopt equation 11 in order to calculate the pressurecorrected rotation curves. For simplicity we will also assume that gas and stars have similar diskscale lengths, equal to the sizes as derived for the star-forming gas from the H α measurements.Note however that there are theoretical reasons why the scale radius of the gaseous disk component,including the part that is not forming stars violently, should be larger than the scale length of thestellar disk (e.g. Sales et al. 2009; Dutton et al. 2010, Guo et al. 2010). This could systematicallybias the derived rotational properties of the disk if the mass fraction of the extended gaseouscomponent is large.
3. Galactic Disk Model
We adopt the model by Mo, Mao & White (1998) of an exponential disk, embedded in a NFW(Navarro et al. 1997) dark matter halo with density distribution ρ DM ( r ) = 4 ρ c ( r/r s )(1 + r/r s ) (12) 6 –where r s is the halo scale radius and ρ c is the dark matter density at r s . The scale radius is relatedto the virial radius r via r s = r /c where c is the halo concentration parameter. The dark halorotation curve corresponding to equation 12 is v DM ( r ) = V (cid:16) r r (cid:17) ln (1 + r/r s ) − ( r/r s ) / (1 + r/r s ) ln (1 + c ) − c/ (1 + c ) . (13)High-resolution numerical Cold dark matter (CDM) simulations (e.g. Zhao et al. 2009) show thatc depends strongly on cosmological redshift. While c decreases with halo virial mass M at lowredshifts, the concentration is roughly constant with c ≈ z ≈ r and M are related to each other through the virial velocity V (Mo, Mao & White 1998) r ( z ) = V ( z )10 H ( z ) , M ( z ) = V ( z )10 GH ( z ) (14) H is the Hubble parameter that depends on cosmological redshift z: H = H (cid:2) Ω Λ + (1 − Ω Λ − Ω M )(1 + z ) + Ω M (1 + z ) (cid:3) / . (15)We adopt a standard ΛCDM cosmology with H = 73 km/s/Mpc, Ω M = 0.238 and Ω Λ = 0 . r = 0, Σ , is determined by the total disk mass M d = m d × M with m d thedisk mass fraction of the galaxy Σ = m d M πr d . (16)The circular velocity curve of an exponential disk is (Freeman 1970) v disk ( r ) = 4 πG Σ r d y [ I ( y ) K ( y ) − I ( y ) K ( y )] (17)with y = r/ (2 r d ) and the I n and K n denoting the modified Bessel functions (Binney & Tremaine2008).We will neglect a bulge because we are interested in the outer disk parts where the bulgecontribution to the rotation curve is in general negligible. In addition, several of the best resolvedSINS galaxies show no evidence for the presence of a significant bulge component (Genzel et al.2008). In this case and including adiabatic contraction (Blumenthal et al. 1986; Jesseit et al. 2002)of the dark halo, the zero-pressure rotation curve is determined from the implicit equation 7 – v ( r ) = v disk ( r ) + v DM ( r ′ ) (18) r ′ = r (cid:20) r × v disk ( r ) r ′ × v DM ( r ′ ) (cid:21) . (19)Given v ( r ) the pressure-corrected rotation curve v rot ( r ) can be calculated from equation 2 or 11.This is easily done in an iterative process. One first determines the disk rotation curve, neglectingpressure as discussed in MMW. Adopting a value of v max /σ , its maximum rotational velocityprovides a first guess for σ which leads to a revised rotation curve and a corresponding new value ofv max and σ . We find that this procedure converges quickly after 10-20 iterations. In the followingwe will call v rot ( r ) the pressure corrected MMW rotation curve. It can be compared directlywith observations (section 4.2). In addition we can calculate the total disk angular momentum J d = 2 π R Σ v rot rdr that will be used in the next section in order to derive the disk spin parameter λ d .
4. Angular Momentum and Baryon Content of High-Z Galaxies
Figure 1 shows the half-light radii r / of the SINS high-redshift galaxies versus their maximumrotational velocity v max . The data points and potential uncertainties are discussed in F¨orster-Schreiber et al. (2009), Cresci et al. (2009) and Law et al. (2009). The errors in r / and v max are of order 1-2 kpc and 20-30 km/s, respectively. We take r / instead of the exponential diskscale length as it is independent of any assumption about the light profile. v max is in general agood approximation of the disk’s rotational velocities outside of r / . The SINS galaxies segregatestrongly into two distinct classes at a critical value of v max /σ ≈
3. We therefore empirically define dispersion-dominated galaxies (open triangles and stars in figure 1) as objects with v max /σ ≤ rotation-dominated galaxies (filled triangles), defined by v max /σ > σ refers to the intrinsic velocity dispersion in the disk, notto the observed line-of-sight or galaxy-integrated dispersion. Figure 1 shows that most of thedispersion-dominated galaxies have radii of order 1-3 kpc while the radii of rotationally dominatedgalaxies are on average a factor of 2-3 larger. In addition, the dispersion-dominated systems haverotational velocities of order 100 km/s while rotation-dominated galaxies rotate with 250 km/s.The specific angular momentum of a dark halo is usually specified by the dimensionless spinparameter (Bullock et al. 2001; Burkert 2009) λ = J √ M V r (20)where J is the total angular momentum of the halo. λ follows a log-normal distribution with amedian of λ = 0 .
035 and a dispersion of 0.55 (Bullock et al. 2001; Hetznecker & Burkert 2006). 8 –Cosmological simulations indicate that in the early phases of protogalactic collapse the gas and darkmatter are well mixed, acquiring similar specific angular momenta (Peebles 1969; Fall & Efstathiou1980; White 1984). If angular momentum were conserved during gas infall and all the gas wouldsettle into the disk, the resulting disk’s specific angular momentum J d /M d would be similar tothe specific angular momentum of the surrounding dark halo. We cannot measure λ directly, butinstead can estimate the disk spin parameter, defined as λ d = J d √ M d V r = λ j d m d (21)with j d ≡ J d /J . If the specific angular momentum of the infalling gas and the resulting disk isequal to the dark halo’s specific angular momentum it follows that λ d = λ .Numerical simulations of galaxy formation find substantial angular momentum loss of theinfalling gas component (for a review see e.g. Burkert, 2009). Its origin is not completely clearup to now and might be attributed to numerical problems or missing physics (for a review see e.g.Mayer et al. 2008). If the numerical calculations are however correct, galactic disks should havespecific angular momenta and values of λ d that are smaller than those of dark matter halos, i.e. onaverage λ d ≤ . λ d , m d and v max /σ . We start with a first guess of thedark matter virial mass (typically M = 10 M ⊙ ). Given m d and by this M d and assuming a diskscale radius r d and a v max /σ the procedure discussed in the previous section gives the correspondingdisk rotation curve and by this the corresponding λ d . In an additional iterative step r d is now variedtill the required value of λ d is achieved.Red dashed lines in figure 1 show the standard MMW model predictions without correcting forpressure effects for a given disk spin parameter λ d (red labels), adopting a high disk mass fraction,equal to the cosmic baryon fraction (m d = M d /M = 0.2) in the left panel and a low value ofm d = 0 .
05 in the right panel. r / is determined from the known disk scale length: r / = 1 . × r d .Stars and open triangles correspond to dispersion-dominated systems, filled triangles to rotation-dominated galaxies. Here we assume that the observed half-light radius, traced by H α , is similarto the half-mass radius of the disk. For m d = 0 . λ d ≈ . − . λ d ≤ . d = 0 .
05 improves the situation considerably. The reddashed lines in the right panel of figure 1 show that, now, MMW models with λ d ≈ . − .
07 fiteven the fast rotators, consistent with the upper half of the dark halo λ distribution.The observed baryonic disk masses provide an additional constraint for theoretical models. Thesymbols in figure 2 show the sum of the stellar mass (from spectral energy distribution analysis)and gas mass (from an application of the Kennicutt-Schmidt star formation relation) of our galaxysample, plotted as function of v max . McGaugh et al. (2000) and McGaugh (2005) find a remarkably 9 –tight correlation between baryonic mass and the circular velocity v circ at a radius where the rotationcurve becomes flat (baryonic Tully Fisher relation): M d /M ⊙ = 50 × ( v circ / ( km/s )) . Interestingly,the high-redshift data shown in figure 2 deviates strongly from this correlation. The red dashedcurves show the expected correlation between disk mass and v max for different values of λ d and forlarge (m d = 0 .
2, left panel) and small (m d = 0 .
05, right panel) disk mass fractions according to theMMW model, neglecting pressure effects. According to figure 1, spin parameters of λ d ≈ .
05 arerequired for m d = 0 .
05. The right panel of figure 2 however demonstrates that these values are notconsistent with the observed disk masses. On the other hand, the left panels of figure 1 and figure 2show that large disk mass fractions of m d = 0 . λ d ≈ . − . λ d ≤ λ , asdiscussed earlier. In section 2 we demonstrated that pressure gradients can significantly affect the rotation curvesof galaxies when the ratio or rotational velocity to velocity dispersion, characterized e.g. by v max /σ ,is sufficiently small. In order to compare the theoretical model with observations we will use themaximum velocity instead of an average velocity, e.g. v . , measured at 2.15 disk scale lengths.In general this could be dangerous as v max could occur anywhere in the disk at radii that are notobserved. However, as demonstrated by figure 3 for the case of BzK-15504, we find that the rotationcurves in general show an extended flat plateau with the maximum at 1-2 disk scale lengths whichis in the observed radius regime. In this case, v max is a good approximation of the typical velocitywithin the flat part of the rotation curve.The blue dotted lines in figure 1 show the correlation between r / and v max for a MMWmodel with pressure correction (equation 11), assuming v max /σ = 2 which is consistent with thedispersion-dominated galaxy sample. The rotation curves are calculated by adopting an exponentialdisk with a given half-light radius r / = 1 . × r d (equation 10) and then calculating iteratively thecorresponding rotation curve as discussed in section 3. Note that v max is now the maximum of thepressure corrected rotation curve which is smaller than the value, neglecting pressure effects. Withpressure correction most of the pressure-dominated galaxies lie in the regime 0 . ≤ λ d ≤ . d = 0 . . ≤ λ d ≤ .
03 for m d = 0 .
05 which is in good agreement with theoreticalexpectations.A significant pressure contribution reduces significantly v max for a given disk mass M d . Theblue dotted lines in figure 2 demonstrate this effect. Like in figure 1, they correspond to disks with v max /σ = 2. Now, the observed masses of pressure-supported SINS galaxies, represented by opentriangles, are consistent with spin parameters λ ≈ . − .
1, independent of the adopted value ofm d . 10 –The stars in figure 2 show galaxies with v max /σ ≤ .
5. These galaxies are characterised byexceptionally low values of v max ≤
100 km/s despite their large masses of M d ≈ M ⊙ asexpected from equation (11) due to the large pressure contribution. The rotation-dominated SINS sample is characterized by average values of v max /σ ≈
5. Theproblem of unusually high spin parameters and baryon fractions therefore cannot be solved byconsideration of pressure gradients. A typical representative of this group is BzK-15504 which hasbeen observed with high angular resolution (Genzel et al. 2006). BzK-15504 is an actively star-forming z = 2 . . +2 . − . × M ⊙ , adopting a Chabrier (2003)initial mass function and a gas mass that varies between 2 . ± . × M ⊙ and 4 . ± . × M ⊙ , depending on the extinction correction applied to the H α line luminosity (F¨orster-Schreiberet al. 2009). The total disk mass is then M d ≈ . × M ⊙ . The radial H α gas surface brightnessand the rest-frame optical stellar light distributions are both consistent with an exponential profilewith scale length of 4.1 kpc.The observed line width indicates irregular gas motions of σ ≈ ±
20 km/s that are constantthroughout the disk outside of the central 3 kpc where a bar and an active galactic nucleus (AGN)strongly affect the gas kinematics. The maximum rotational velocity is v max = 258 ±
25 km/s,so that v max /σ = 5.7. The dark matter virial mass is not well constrained. A minimum valuecan be inferred if one adopts a disk mass fraction, close to the cosmic baryon fraction f b = 0 . M ≥ M d /f b ≈ × M ⊙ .The dotted red and blue curves in the left panel of figure 3 show the disk and dark matterrotation curves, respectively, adopting a MMW model with the above mentioned disk parametersand M = 8 × M ⊙ . Both components are equally important with peak rotational velocitiesat r ≈
10 kpc of 250 km/s and 280 km/s. The upper black curve shows the resulting combinedrotation curve, neglecting pressure effects. It peaks at 370 km/s and is clearly inconsistent withthe observations (red open circles) that peak at ∼
260 km/s. Including turbulent pressure with σ = 45 km/s does not significantly change the rotation curve (black points). A test calculation,adopting σ = 45 km/s shows that increasing M (decreasing m d ) makes the problem even worsewhile decreasing M would require that the galxy has a baryon fraction larger that the cosmicvalue.Is this disagreement an observational problem? The uncertainty in the measured disk velocitydispersion is large, of order 20 km/s. However a dispersion of even 65 km/s (dashed black curves infigure 3) makes no big difference. The uncertainties in the determination of the rotation curve areof order 25 km/s, too small compared with the disagreement of almost 100 km/s. The only possiblesolution appears to be a strong reduction of either the baryonic or the dark matter mass withinthe inner 10 kpc. A test calculation shows that the baryonic disk mass would have to be reduced 11 –by a factor of 2 to ∼ × M ⊙ for a pressure corrected MMW model with velocity dispersionof 65 km/s to fit the observations. The estimates of the stellar and gas masses are subject tomany uncertainties and systematics. Spectral energy distribution fitting for BzK-15504 using theMaraston (2005) models yields stellar masses of M ∗ = 9 . . / − . × M ⊙ and gas masses of M gas = 3 . . / − . × M ⊙ using the Schmidt-Kennicutt relation from Bouch´e et al. (2007)and the same extinction towards HII regions as towards the stars. The gas masses would be a factorof 2 higher if the extinction towards HII regions is a factor of 2 higher than towards stars (Calzettiet al. 2004). The best-fit Bruzual & Charlot (2003) model gives M ∗ = 10 . . / − . × M ⊙ and M gas = 2 . . / − . × M ⊙ . Unless the stellar initial mass function is bottom-light ortop-heavy it therefore seems difficult to decrease substantially the baryonic disk mass of the galaxy.Another possibility is a smaller dark halo mass in the disk region. One critical assumptionthat enters the MMW model is that the dark halo reacts to the formation of the galactic diskby contracting adiabatically. The right panel of figure 3 shows the situation for a MMW model,neglecting dark halo contraction. A comparison with the left panel demonstrates the strong effectof adiabatic contraction. Although the dark matter virial parameters in both cases are the same,without adiabatic contraction the contribution of the dark halo (blue dashed line) within the diskregion is small, leading to much better agreement of the model with the observations, especially ifwe take into account a turbulent pressure, corresponding to a velocity dispersion of σ = 65 km/s(upper dashed black line), that is still within the observed uncertainties and expected if the diskvelocity dispersion tensor would be anisotropic (Aumer et al. 2010).The problem discussed for BzK 15504 exists for all rotation-dominated galaxies that are rep-resented in the figures 1 and 2 by black filled triangles. Due to their large values of v max /σ ≈ λ d = 0 . d . The solid black lines in both figures showthe strong effect of neglecting adiabatic dark halo contraction. According to figure 1, the observeddisk radi require λ d ≈ .
07 for m d = 0 . λ d = 0 .
035 for m d = 0 .
05. Figure 2 shows that thesespin values are also consistent with the observed disk masses.
5. Origin of Gas Turbulence in High-Redshift Disk Galaxies
The MMW models also provide insight into the origin of the observed gas turbulence. Letus propose that the main driver of clumpiness and turbulence in gas-rich high-redshift disks isgravitational disk instability. Then we expect gas-rich disks to stay close to the gravitationalstability line because of the following reason. A disk that is kinematically too cold with smallvelocity dispersions is highly gravitationally unstable. Gravitational instabilites generate densityand velocity irregularities that drive turbulence and heat the system kinematically. As a result,the gas velocity dispersion increases till it approaches the stability limit where kinetic driving bygravitational instabilities saturates. A disk with even higher velocity dispersions would be stable. 12 –Here the turbulent energy would dissipate efficiently and the velocity dispersion would decreaseagain until it crosses the critical velocity dispersion limit where gravitational instabilities becomeefficient again in driving turbulent motions. In summary, galactic disks should settle close to thegravitational stability line that is determined by the Toomre criterion (Toomre 1964; Wang & Silk1994) Q ≡ κπG (cid:18) Σ g σ g + Σ ∗ σ ∗ (cid:19) − ≤ Q c . (22) κ is the epicyclic frequency that is related to the local angular circular velocity Ω at radius rthrough κ = rd Ω /dr + 4Ω and Q c is the critical value which is of order unity (Goldreich &Lynden-Bell 1965; Dekel et al. 2009b). Σ ∗ and σ ∗ are the stellar surface density and velocitydispersion, respectively. As a test, let us focus again on BzK 15504. As most of the stars in BzK15504 are likely to have formed during the presently observed star burst we can assume that thestellar velocity dispersion is similar to the observed turbulent gas velocity, i.e. σ ∗ ≈ σ g . In addition,the observations show that both components have similar exponential disk scale lengths. If the diskis close to the instability line, its turbulent gas velocity dispersion at any point r is then σ = πG Σ κ (23)where Σ( r ) is the local baryonic (gas+stars) disk surface density. The lower black dot-dashedcurve in the right panel of figure 3 shows the predicted gas velocity dispersion for BzK-15504. It isindeed almost independent of radius and within the uncertainty in agreement with the observed,radially constant value of 45 ±
25 km/s. We thus conclude that BzK-15504 is a marginally unstablestar-forming disk (Genzel et al. 2006), driven by gravitational instabilities.We calculated the velocity dispersion profile for all SINS galaxies with v max /σ ≥ d was constrained by fitting the observed maximum velocity of the galaxies.In all cases the theoretically derived velocity dispersion σ theo , adopting equation 23, is almostconstant within 1 and 2 disk scale radii. Figure 4 compares the average value of σ theo within 1and 2 r d with the observed velocity dispersion σ obs . Dispersion-dominated (open triangles) androtation-dominated (filled triangles) systems have similar gas velocity dispersions, indicating thatthe difference in v max /σ is due to a difference in rotational velocities and not a result of differencesin the turbulent gas velocity. Despite the large uncertainties, the theoretical and observed velocitydispersions agree well, strengthening the suggestion that gravitational instabilities are the majordriver of turbulence in high-redshift star-forming galaxies. 13 –
6. Summary and Discussion
We have shown that pressure gradients in turbulent galactic disks can significantly affect theirrotation curves. This effect is well-known for thick stellar disks, leading e.g. to an asymmetric driftof kinematically hot stellar populations in the Galaxy (Binney & Tremaine 2008). A similar effectis found in models of dust growth in protoplanetary disks where dust particles on ballistic orbitsrotate faster than the disk gas which rotates sub-Keplerian due to pressure gradients, leading tofatal dust migration into the central star (e.g. Takeuchi & Artymowicz 2001).We analysed the SINS sample of dispersion-dominated high-redshift star-forming galaxies.Including pressure effects and adopting an exponential gas disk with scale length similar to thestellar disk, the models can explain the properties of the dispersion-dominated SINS galaxies verywell, with disk spin parameters of λ d ≈ . − .
05 and disk mass fractions of m d ≈ . − . v max /σ ≥
3, pressure gradients cannot stronglyaffect the rotational velocities. Analysing as a test case the galaxy BzK 15504 we showed that itsrotation curve allows no significant contribution of dark matter within the visible disk region. Thiscan be achieved with a standard NFW halo that did not contract adiabatically in response to theformation of the galactic disk. The MMW models of the rotation-dominated sample, neglectingadiabatic dark halo contraction, lead to reasonable values of λ d ≈ .
035 if the disk mass fractionis low (m d ≈ .
05) while in the case of high mass fraction (m d ≈ . λ d ≈ . λ d ≥ . λ d values then their dark matterhalos. One of the most promising suggestions are selective galactic outflows of especially low-angular momentum gas from galactic centers (e.g. Maller & Dekel 2002; Dutton & van den Bosch2009).Sales et al. (09) compared MMW model predictions with the SINS high-redshift disk galaxiesand found a good agreement in the rotational velocity versus disk scale length plane (similar to Fig.1) for reasonable spin parameters λ d ≈ . − .
06. They assumed values of m d = 0 .
05 and includeddark halo contraction. Unfortunately, Sales et al. do not compare their predicted disk masses withobservations. In addition, they argue that the observed gas disk radii are a factor κ r = 1 . λ d by this factor in order to compare their model with observations.This implicitly assumes that the gas component has a negligible mass as otherwise it would affect 14 – λ d . The assumption of a low gas-to-star fraction is not supported by the SINS observations whichtypically indicate disk gas fraction of order 30-60% (Tacconi et al. 2010). In addition, for thosecases where stellar data is also available Cresci et al. (2009) find similar scale radii for the stars andgas with κ r ≈
1, justifying our assumption. Still, more complex multi-component MMW modelsmight be interesting in order to better understand the physical properties of systems with stellarand gaseous disks that are characterised by different scale radii.The problem that adiabatic halo contraction leads to compact galactic disks with scalingrelations that are not in agreement with observations has been discussed for low-redshift galaxiese.g. by Dutton et al. (2007, with references therein) who advocate a model in which the dark haloactually expands rather than contracting. We find that this effect is important also for high-redshiftgalaxies. Several solutions are currently being discussed. Gnedin et al. (2004) and Gustafssonet al. (2006) argue that the circular orbit adiabatic contraction model (Barnes & White 1984)considerably overestimates the amount of dark matter contraction. The numerical simulations ofJesseit et al. (2002) however find good agreement with the analytical expression. The dark mattermass fraction in the disk region could also be reduced if one assumes cored dark matter halos(Burkert 1995; Salucci & Burkert 2000), resulting e.g. from dark matter annihilation, dark matterparticle scattering or dynamical interaction. Halo expansion could be triggered by dynamicalinteraction with massive subclumps or molecular clouds in the disk (El-Zant et al. 2001; Duttonet al. 2007, Mashchenko et al. 2006; Johansson et al. 2009, Abadi et al. 2010; Jardel & Sellwood2009) or with bars (e.g. Weinberg & Katz 2002; Sellwood 2008). Rapid outflows of gas would alsolead to halo expansion (Navarro, Eke & Frenk 1996; Gnedin & Zhao 2002; Read & Gilmore 2005).Whatever the origin, our result demonstrates that the problem of inefficient dark halo contrac-tion is related to the earliest phases of galaxy formation. Although not required in order to producereasonable spin parameters and disk mass fractions, the processes that suppressed adiabatic halocontraction in rotation-dominated galaxies might also have been active in dispersion-dominatedsystems. In this case, most dispersion-dominated systems would be characterized by even smallerspin parameters of λ d = 0 . − . Acknowledgments:
A.B. is partly supported by a Max-Planck-Fellowship. This work was sup-ported by the DFG Cluster of Excellence ”Origin and Structure of the Universe”. The DarkCosmology Centre is funded by the Danish National Research Foundation. We thank A. Dekel, J.Navarro and L. Sales for useful discussions. We thank the referee for a detailed and constructivereport that greatly improved the quality of our paper.
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This preprint was prepared with the AAS L A TEX macros v5.2.
20 –Fig. 1.— The left and right panels show the disk half-light radii r / versus the maximum ro-tational velocities v max of models with disk mass fractions of m d = 0 . d = 0 .
05, respec-tively and compare them with the SINS high-redshift disk sample. Open triangles correspond todispersion-dominated galaxies, filled triangles to rotation-dominated objects. Stars show extremelydispersion-dominated systems with v max /σ ≤ .
5. Red dashed lines show the theoretically pre-dicted correlation between r / and v max if pressure effects are neglected for various values of thedisk spin parameter λ d (red labels) Blue dotted lines show MMW models including the effect ofa pressure gradient and adopting v max /σ = 2. The black solid lines represent rotation-dominatedgalaxies with v max /σ = 5, neglecting adiabatic dark halo contraction. 21 –
50 100 150 200 250 30050 100 150 200 250 300
Fig. 2.— Open and filled triangles show the observationally inferred disk masses versus themaximum velocity of dispersion-dominated and rotation-dominated high-redshift SINS galaxies,respectively. The stars show extremely dispersion-dominated galaxies with v max /σ ≤ .
5. The reddashed curves show the predictions of the standard MMW model, adopting a disk mass fraction ofm d = 0 . d = 0 .
05 in the right panel. Red labels indicate the correspondingdisk λ d parameters. The blue dotted lines show the situation if turbulent pressure is taken intoaccount, adopting v max /σ = 2. Black, solid curves correspond to MMW models with v max /σ = 5,neglecting adiabatic contraction of the dark halo. 22 –Fig. 3.— Open red circles in both panels show the inclination and resolution corrected intrinsicrotation curve of BzK-15504, inferred from fitting the observed two-dimensional distribution of H α velocities (Genzel et al. 2006). We focus on the rotational properties outside of 3 kpc as the innerregions are affected by the central AGN and bar. The dotted red and blue lines in the left panelshow the theoretically expected contribution of the disk and dark halo component, respectively,adopting a cosmic baryon fraction m d = f b = 0.2 and an adiabatically contracted dark halo withconcentraction c=4. The combination of both curves leads to the zero-pressure total rotation curve(upper black line) that exceeds the observed maximum rotational velocity by more than 100 km/s.The black points and the dashed black curve correspond to the pressure corrected rotation curve,including pressure effects with a gas velocity dispersion of 45 km/s and 65 km/s, respectively.The right panel shows the situation without adiabatic dark halo contraction. Symbols and linesare the same as in the left panel. Now the resulting rotation curve is in much better agreementwith the observations. The lower dot-dashed black curve shows the theoretically predicted velocitydispersion profile assuming a constant Toomre stability parameter of Q=1. 23 –Fig. 4.— The observed velocity dispersion σ obs of the SINS high-redshift galaxy sample is comparedwith theoretical expectations σ theo , adopting a pressure-corrected MMW model without adiabatichalo contraction. σ theotheo