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

2. 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

3. 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.

4. 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

5. 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 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

6. 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

7. 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

8. 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.

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

10. 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

11. 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).

12. 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.

13. 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 ?

14. 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)

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

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

17. 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:

18. 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
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

19. 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
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
o - 
 - MB
The surface densities
of dark halos ~ o rc
are almost independent
of luminosity

20. 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 = ± 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

21. 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.

22. 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 - 

23. The required shifts are similar for the dSph and dIrr
shift in MB is mag
shift in rc is dex
shift in  is 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

24. 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 M
This halo would have a mass around 10^9 M_sun .

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

26. 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 ~ M,
then for Carina with  ~ 20 km/s, M ~ 109 M

27. 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

28. 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