Scaling laws for dark halos of galaxies Ken Freeman Australian National University NGC 628 - large spiral GR8 - small dwarf Dark Matters: Paris, Dec 2017

The disks of spiral galaxies lie in a dark matter halo. In the brighter spirals, the baryons are mostly self-gravitating, but their self-gravity is weaker in the fainter systems. NGC 628 The faint dwarfs (MB > -13) are strongly baryon-depleted. Their M/L ratios are ~ 100 to 1000, and the self-gravity of the baryons is negligible. We can regard the baryons as test particles in the gravitational field of the dark halos. Carina

Dark Halo Scaling Laws How do the properties of dark halos change with the brightness MB (or circular velocities Vc ) of the galaxies ? Galaxies of different masses form at different redshifts. Sample includes rotationally supported Sc-Im disk galaxies and faint pressure-supported dwarfs: MB = -2 to -22 For the late type (Sc-Im) disk galaxies, we use rotation curves to measure the central density o and the core radius rc of the dark halos from rotation curve decomposition. For the dwarfs, we use Jeans equation (hydrostatic equilibrium) models to get the central density o from their velocity dispersion and baryon distribution Discussion follows Kormendy & Freeman (2016) Galaxies of different masses form at different epochs - dwarfs very early z ~ 20, large spirals later z ~ 3.

Dark halo models What is the distribution of matter in dark halos ? • The NFW cusped dark halo distribution (from N-body simulations) • Distributions with flat cores: the pseudo-isothermal sphere (widely used by observers) and the non-singular isothermal sphere (King model) and the Burkert model

N-body simulations favor the “cosmologically motivated” cusped halos, while observations indicates that the dark halos of real galaxies have near-constant density cores. • rotation curve studies of low surface brightness galaxies (e.g. Oh et al 2011) • dynamics of dwarf spheroidal galaxies (e.g. Salucci et al 2012; Scl: Agnello & Evans 2012. Fornax: Amorisco et al 2013). N-body DM-only simulations and observations probably both right baryon loss heats cusps We use the pseudo isothermal sphere model in the form scaled approximately to the King core radius rc

Measuring o and rc for dark halos of spiral galaxies Calculate shape of rotation curve for disk from its light distribution. Adopt maximum disk: Milky Way and nearby large spirals are maximal. Add rotation curve for gas and dark halo: fit o and rc to observed rotation curve. maximal Know shape of disk V(R) but not amplitude - degenerate - need some way to get V(R) for disk and hence V(R) for halo Old problem; recent work favors maximal disks for bright spirals. halo Use this method for galaxies with MB > -14 Fainter galaxies are not rotationally supported. disk van Albada et al 1985

Maximal Disks • Aniyan 2017 o Bottema 1993 • DiskMass 2011 DMS data and Bottema (1993) show sigma_LOS. Shaded region is for maximal disk. Our points show sigma_z_hot, and lie in the ‘maximal’ region. • Aniyan 2017 o Bottema 1993 • DiskMass 2011 adapted from Bershady et al. 2011 7

Measuring o for dark halos of faint dwarfs (MB > -13, slow rotation) These systems are supported against gravity by their stellar velocity dispersion (dSph) or turbulent gas pressure (dIrr). For dIrr galaxies, use the distribution and kinematics of HI. For dSph, use distribution and kinematics of stars as tracers of the equilibrium. The velocity dispersions in dSph and dIrr galaxies appear to be isothermal, with dispersions  between about 5 and 10 km/s.

Stellar velocity dispersion for dSph galaxies (Walker et al 2009) Roughly isothermal, 5-10 km/s  (km/s)

Less data on the radial velocity dispersion profiles for HI in faint dIrr galaxies. Walter et al 2008 (THINGS) find near-constant velocity dispersions for most of their fainter galaxies (DDO154, Ho I, M81 DwA, DDO 53) GR8 - Carignan et al 1990 Sextans A - Skillman et al 1988

Need a model for the baryon distribution in order to estimate the halo central density o from the hydrostatic equilibrium of dwarfs. Simple ad hoc models like King models or Plummer models are often used. What would we expect ? Simulations of subhalo evolution indicate that the halos of dwarf galaxies are more extended than their baryons, and have larger circular velocities. Assume that the baryons inhabit the core region of their dark halos. The velocity dispersion of stars in dSph and HI in faint dIrr are roughly isothermal. Assume they are isotropic and spherical. Calculate the equilibrium density distribution of an isothermal, isotropic, spherically symmetric tracer population in the uniform density core of their dark halo. Turns out to be a gaussian volume density distribution (  also gaussian surface density distribution).

the spherical Jeans equation is For isotropic isothermal test particles in core of DM halo of density o, the spherical Jeans equation is tracer dark halo and the expected density *(r) and the surface density *(R) of the test particles are both gaussian where the density o of the halo core, the velocity dispersion * and scale length a of the test particles are related by If the observed density profiles are gaussian, then the measured * and a values for the baryons give the density o of the dark halo.

If the baryons (e.g. HI in a dIrr galaxy) has solid body rotation with angular velocity *, then the baryon surface density distribution is still expected to be gaussian, and the central density o of the halo is For this study, we avoided galaxies for which the rotation correction was more than ~ 30 percent. Solid body rotation is common in dIrr galaxies, so one could probably use more of these systems. Do real dwarfs have gaussian surface density distributions ?

Gaussian fits dSph: star counts dIrr: HI surface density DDO 210 is Aquarius dwarf ! (l,b) = (34,-31), vel = -141, distance 1.0 Mpc Plummer model is just soften point mass potential  ~ r -5 gives very steep pressure gradient best Plummer model (Plummer models are often used for dwarf galaxies)

Gaussian fits to starcount ultrafaint dSph Gaussian fits to starcount surface density for 11 dSph galaxies

Gaussian fits to HI surface density distributions for 12 dIrr galaxies (all several times larger than the HI beam)

The observed gaussian surface density distributions for the baryons in faint dSph and dIrr galaxies show that our assumptions are at least roughly consistent: DM halo with constant density core isothermal, isotropic tracer spherical approximation From the hydrostatics, we can only estimate the DM central density o for the dwarfs. The DM core radius cannot be estimated unless the baryons extend out beyond the rc of the halo. ------------------------------------------- Now look at the scaling relations of o, rc and the halo velocity dispersion  with MB. The dispersion is not independent:

are almost independent rc - MB o - MB Scaling laws for dark halos of spirals (o, rc ) vs MB Halos of brighter galaxies have lower density o  LB - 0.39 rc - MB Halos of brighter galaxies have larger core radii This is why MOND works - (G * surface density) is an acceleration and is almost constant. rc  LB+ 0.44 The surface densities of dark halos ~ o rc are almost independent of luminosity

Halos of brighter galaxies have lower density Scaling laws for dark halos of spirals (o, rc, ) vs MB o - MB o - rc Halos of brighter galaxies have lower density  - rc rc - MB o  LB - 0.39 Halos of brighter galaxies have larger core radii This slide shows scaling laws of each parameter against the others. From ISO scaling, expect covar (rho_0).(r_c^2) = const at const sigma rc  LB + 0.44 o -   - MB The surface densities of dark halos ~ o rc are almost independent of luminosity

Good agreement with CDM ** Power spectrum of initial density fluctuations: | k|2  kn Predicted Scaling* Observed Scaling n -1.83 ± 0.19 -2.07 ± 0.08 -2.08 ± 0.18 average n = -1.99 ± 0.10 Good agreement with CDM ** DM parameter correlations provide a measure of n on smaller mass scales than are accessible to most other techniques This is an important consistency check. If our decompositions were very wrong (e.g. if disks were very submaximal), then this check would fail. * Peebles 1974; Gott & Rees (1975); Djorgovski 1992 ** Shapiro & Iliev 2002

Now add the dwarfs to the scaling relations: recall that we have measured o for the halos of the dwarfs, but we cannot measure their halo core radius rc or their halo velocity dispersion . Plot their halo o and their observed baryon rc (derived from the gaussian a-parameter) and baryon . Expect that • our halo densities o for the dwarfs are roughly right • baryon rc is less than the DM rc • baryon  likely to be less than the DM  • dwarfs have probably lost more baryons than the spirals, so are likely to be fainter in the MB scaling laws.

plotted values of rc and  are for their baryons. Now add the faintest dwarfs. We have measured o for their halos and they are plotted. The plotted values of rc and  are for their baryons. The dwarfs with MB > -13 have lost most of their baryons - the dispersion  and rc for their halos is probably larger than for the baryons. The dwarf halo densities overlap with the faintest spirals. Assume that the dwarf halos lie on the same scaling relations as the spirals. Shift the dwarf halos in MB, rc and  so that they lie on the spiral scaling relations. o - rc o - MB Assume that our halo densities are right, so no shifts in density - red labels o - 

The required shifts are similar for the dSph and dIrr shift in MB is -4.0 -3.5 mag shift in rc is 0.70 0.85 dex shift in  is 0.40 0.50 dex The shifts represent • loss in baryons (factor 30 relative to the spirals) - consistent with their M/L • difference in rc between the baryons and the dark halos of the dwarfs (factor 5) • difference in  between the baryons and the dark matter (factor 3) Remember we are looking at the very faintest of the dwarfs here. Difference in sigma is measure of difference in circular velocity

If this is all correct, then ... the stars in a dSph like Carina, with The difference in rc between the baryons and the dark halos of the dwarfs is a factor 5 The difference in  between the baryons and the dark matter is a factor 3 If this is all correct, then ... the stars in a dSph like Carina, with a baryonic core radius of 200 pc and a velocity dispersion of 7 km/s, are immersed in a dark halo with a core radius of about 1 kpc and a velocity dispersion of 20 km/s (or circular velocity of 30 km/s). Consistent with assumption that the luminous dwarf lies within the flat core of its dark halo. Compare with large spiral M101 ... This halo would have a mass around 10^9 M_sun .

M101 - circle shows its dark halo core radius (19 kpc) Two more points ..

The constant central surface density of the dark halos: Kormendy & Freeman (2004, 2016). Seen more directly by Donato et al (2010) and Salucci et al (2012) Constant surface density means constant acceleration over whole range of MB. That is the MOND acceleration. The dwarfs are more massive than one might think from their small baryonic velocity dispersions. May help for direct detection. The constant surface density of the halos means that the halos follow a Faber-Jackson law M ~  4. e.g. for the MW halo,  ~ 150 km/s and M ~ 2.1012 M, then for Carina with  ~ 20 km/s, M ~ 109 M

Galaxies are “dim” if Vcirc < 40 km s-1,  < 30 km s-1 The typical dSph galaxy has visible matter  = 10 km s-1 and DM Vcirc ≈ 40 km s-1 and is almost completely dark (M/L ~ 100). As L decreases, dwarfs become more numerous and more dominated by DM. There may be a population of dwarfs with Vcirc < 40 km s-1 that are completely dark: they were unable to retain their baryons or maybe never acquired them V_circ is for everything : baryons + DM - dominated by DM V_circ comes from sigma_DM Undiscovered dark dwarfs could solve the old problem that CDM predicts too many dwarf galaxies

Conclusions All the best to Joe • Pressure-supported dwarfs have the gaussian density distributions expected if their baryons are isothermal, isotropic, and lie within the core of their ~ isothermal sphere halos. • Dark halos have well defined scaling laws extending over about 10 mag for the rotationally supported disk galaxies, and about 20 mag if the pressure-supported dwarfs are included. The slopes of the scaling laws imply n = -2 for the fluctuation spectrum: this is a consistency check on the adoption of maximum disks. • The DM correlations provide an estimate of (a) the baryon deficiency in dwarf galaxies relative to the larger disk galaxies and (b) the dynamical parameters rc and  for the dark halos of the dwarfs. • There may be a population of dark dwarfs with Vc < 40 km/s All the best to Joe