galaxies: structure, dynamics, and...

Elliptical Galaxies (II)

Galaxies: Structure, Dynamics, and Evolution

Layout of the CourseFeb 5: Review: Galaxies and CosmologyFeb 12: Review: Disk Galaxies and Galaxy Formation BasicsFeb 19: Disk Galaxies (I)Feb 26: Disk Galaxies (II)Mar 5: Disk Galaxies (III) / Review: Vlasov EquationsMar 12: Review: Solving Vlasov EquationsMar 19: Elliptical Galaxies (I)Mar 26: Elliptical Galaxies (II)Apr 2: (No Class)Apr 9: Dark Matter HalosApr 16: Large Scale StructureApr 23: (No Class)Apr 30: Analysis of Galaxy Stellar PopulationsMay 7: Lessons from Large Galaxy Samples at z<0.2May 14: (No Class)May 21: Evolution of Galaxies with RedshiftMay 28: Galaxy Evolution at z>1.5 / Review for Final ExamJune 4: Final Exam

this lecture

First, let’s review the important material from last week

What is the nature of elliptical galaxies?

While progenitors to elliptical galaxies experienced lots of star formation, elliptical galaxies themselves

experience almost no star formation

End state of galaxy formation!

Dominated by the random motions of its component parts (stars,

hot gas, dark matter)

Only way for an elliptical galaxy to transform into another type of galaxy (e.g., spiral) is if lots of cold gas

cools onto it (but this may not happen!)

What we can learn from Images of Elliptical Galaxies?

Important clues can come from looking at the two dimensional surface brightness profiles of elliptical galaxies.

Two Types of Deviations from Elliptical Isophotes are Observed

OutlineDisk galaxies

Elliptical galaxies

Luminosity distributionsShells and ripplesColor gradients

We will now look at fits in a boxy galaxy.

Piet van der Kruit, Kapteyn Astronomical Institute Luminosity distributions: Parameters

OutlineDisk galaxies

Elliptical galaxies

Luminosity distributionsShells and ripplesColor gradients

And here are fits a disky galaxy.

Piet van der Kruit, Kapteyn Astronomical Institute Luminosity distributions: Parameters

Boxy Disky

(Few Percent)

What we can learn from Images of Elliptical Galaxies?

In studying the elliptical galaxies, one quantifies the ellipticity and position angle of the galaxy at different surface brightness levels.

It is not uncommon to find that the position angle and ellipticity changes as a function of radius!

This twisting of the isophotes is expected for a triaxial profile from projection

effects. Triaxial Profile

Center of Ellipticals: Core vs. Power-Law Galaxies

Core in centers of ellipticals likely created from Scouring by Super Massive Black Holes (109 solar masses) During Merging Definition of Break radius

I( r) = Ib2(" -# )/$ (rb/r)

# [1+(r/rb)$ ](# -" )/$

rb = break radius where power-lawchanges slope and Ib is the surface brightness at the break

This is a five(!) parameter fit:" is the outer power-law slope# is the inner power law slope

$ defines the sharpness of thetransition

Break radius vs. Absolute magnitude

Shape of Ellipticals:

• Ellipticals are defined by En, where n=10!,

and !=1-b/a is the ellipticity.

• Note this is not intrinsic, it is observer

dependent!


• 3-D shapes – are ellipticals predominantly:

– Oblate: A=B>C (a flying saucer)

– Prolate: A>B=C (a cigar)

– Triaxial A>B>C (a football)

– Note A,B,C are intrinsic axis radii

• Want to derive intrinsic axial ratios from observed

– Can deproject and average over all possible observing angles to do

this

– Find that galaxies are mildly triaxial:

• A:B:C ~ 1:0.95:0.65 (with some dispersion ~0.2)

– Triaxiality is also supported by observations of isophotal twists in

some galaxies (would not see these if oblate or prolate)

Definition of Break radius

I( r) = Ib2(" -# )/$ (rb/r)

# [1+(r/rb)$ ](# -" )/$

rb = break radius where power-lawchanges slope and Ib is the surface brightness at the break

This is a five(!) parameter fit:" is the outer power-law slope# is the inner power law slope

$ defines the sharpness of thetransition

Break radius vs. Absolute magnitude


• Ellipticals are defined by En, where n=10!,

and !=1-b/a is the ellipticity.

• Note this is not intrinsic, it is observer

dependent!


• 3-D shapes – are ellipticals predominantly:

– Oblate: A=B>C (a flying saucer)

– Prolate: A>B=C (a cigar)

– Triaxial A>B>C (a football)

– Note A,B,C are intrinsic axis radii

• Want to derive intrinsic axial ratios from observed

– Can deproject and average over all possible observing angles to do

this

– Find that galaxies are mildly triaxial:

• A:B:C ~ 1:0.95:0.65 (with some dispersion ~0.2)

– Triaxiality is also supported by observations of isophotal twists in

some galaxies (would not see these if oblate or prolate)

Surface Brightness

as super massive black holes sink to the center of new

galaxyformed from the merger of two smaller galaxies

these black holes eject stars that they collide with gravitationally

before merger after merger

Mass of ejected stars is similar to mass of supermassive black hole.

Super Massive Black Holes in Centers of Elliptical Galaxies

MBH/σ relationship

  The mass of the black hole is highly correlated with the luminosity and total mass of the spheroid component (spiral bulge or elliptical)

The mass of the black hole is ~ 108 to 109 solar masses and is strongly correlated with the luminosity and total mass of the elliptical galaxy. This is called the M-σ relation. There are a

number of competing explanations for this trend.

Velocity DispersionLuminosity

Black Hole Mass

One possibility = rotational flattening

Why do Ellipticals Have an Elliptical Morphology?

rotating objects tend to be elongated away from

axis of rotation

this mechanism has an impact on the shape of the planets in the solar

system, especially Jupiter and Saturn

Why do Ellipticals Have an Elliptical Morphology?

Second possibility = anisotropy in the velocity dispersion

consider the formation of elliptical galaxies from the merger of two smaller galaxies

20-3-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c02-11

where M is the mass of the system and v20 is the mass-

weighted mean square rotation speed. In the same way

Πxx = Mσ20 ,

where σ20 is the mass weighted mean square velocity

along the line of sight to the galaxy. We parametrizethe dispersion in the z direction by δ

Πzz = (1 − δ)Πxx = (1 − δ)Mσ20 .

δ < 1 measures the anisotropy of the velocity-dispersiontensor. At δ = 0, the velocity dispersion in the z andx direction is the same, and the system is “isotropic”.For other δ, the system is “anisotropic”. We can write

v20

σ20

= 2(1 − δ)Wxx

Wzz− 2

As it turns out, for galaxies with isodensity surfaceswhich are similar ellipsoids, the term Wxx

Wzzdepends

only on the ellipticity of the ellipsoids. Hence v0σ0

is afunction of ϵ and δ only. If the model is isotropic, v/σshould depend in a very simple way on the observedellipticity. The deviation of the observations from thisrelation tells us how anisotropic the galaxies are.In practice, we also have to take into account projec-tion effects: the observer measures not the intrinsicmotions, but only the motions along the line-of-sight.The figure below shows the model predictions.


The observations indicate that many galaxies are notisotropic - i.e., not flattened by rotation:

Bright elliptical galaxies rotate very slowly, and hencemust be anisotropic to generate their shapes. Fainterellipticals generally rotate faster - and are more or lessconsistent with being isotropic.The conclusion is that the bright elliptical galaxies

Lower Luminosity Spheroids More Luminous Spheroids

How does the predicted relationship between v/σ and ellipticity (ε) compare to that found for elliptical galaxies observed in the real universe?

Oblate

isotro

pic

rotator Obla

te iso

tropic

rotator

It depends on the luminosity of the elliptical galaxy.

From the observed v/σ versus ellipticity (ε) relationship, it seems clear that there is a qualitative difference between the most luminous elliptical

galaxies and lower luminosity elliptical galaxies:

This dichotomy is seen in many of the other properties of elliptical galaxies as well:

Luminosity

Rotation Rate

Flattening

Isophotes

Shape

Profile in Center

Sersic Parameter

X-ray/radio

High Low

Slow Faster

Anisotropy Rotation

Boxy Disky

Triaxial Oblate

Core Power-Law

n>=4 n<=4

Loud Quiet

This suggests that higher luminosity and

lower luminosity elliptical galaxies may

form in different ways!

NOW new material for this week

How the apparent rotational flattening of elliptical galaxies depend upon its isophotal structure (boxy vs. disky)

Kinematics of ellipticals (NGC 1399) Kinematics of Ellipticals:

• Vrot/) correlates with luminosity– Lower luminosity ellipticals have higher Vrot/) -- rotationally

supported

– Higher luminosity ellipticals have lower Vrot/) -- pressure supported

• Vrot/) correlates with boxy/diskiness– Disky ellipticals have higher Vrot/) -- rotationally supported

– Boxy ellipticals have lower Vrot/) -- pressure supported

• Rotation implies that ellipticals are not relaxed

systems– Some have kinematically decoupled cores, or rotation along their

minor axis (implies triaxiality)

Vrot/) vs luminosity Vrot/) & minor axis rotation vs a4/a

disky

boxy

a4 parameter defining “boxiness”or “diskiness”

(v / σ)*

relative to thatrequired for

rotational support

Disky and boxy elliptical isophotes Shape of Ellipticals – disky/boxy

• Disky/boxy correlates with other galaxy

parameters:

– Boxy galaxies more likely to show isophotal twists

(and hence be triaxial)

– Boxy galaxies tend to be more luminous

– Boxy galaxies have strong radio and x-ray

emission

– Boxy galaxies are slow rotators

– In contrast – disky galaxies are midsized ellipticals,

oblate, faster rotators, less luminous x-ray gas

Radio power and x-ray luminosity vs a4/a Kinematics of Ellipticals:

• Measured via the integrated absorption

spectra

• ) is the velocity dispersion (random

motion) observed via width of absorption

line compared to template spectra

• Vr is the mean radial velocity

– Rotation is evident as a gradient in Vr

across the face of the system

How the radio power and x-ray luminosity of elliptical galaxies depend upon its isophotal structure (boxy vs. disky)

Part of the above trend may be due to the fact that boxy galaxies are more luminous overall -- and larger galaxies are also expected to brighter at x-

ray and radio wavelengths

We can also see this dichotomy between the properties luminous ellipticals and lower luminosity ellipticals

in the radius vs. luminosity relation:

Elliptical galaxies:

•! There are other correlations

–! Brighter ellipticals are bigger

–! Brighter ellipticals have lower average surface brightness

–! Can put the above two together to form the Kormendy

relation – larger galaxies have lower surface brightnesses --

µB,e = 3.02 log re + 19.74

–! Brighter ellipticals have lower central surface brightness

–! Brighter ellipticals have larger core radii -- the core radius is

the radius where the SB drops to ! that of the central SB,

I(r=0)

Effective radius vs Absolute magnitude

Average surface brightness vs Absolute magnitude Kormendy relation (1977)

Optical Luminosity BrightFaint

Half-LightRadius

How might these two classes of elliptical galaxies arise?

Case #1: “Wet” Mergers

Galaxy(with gas)

Galaxy(with gas)

Resulting in Disky Ellipticals?

(tends to occur more frequently for lower mass galaxies, when galaxy evolution less

advanced)

Case #2: “Dry” Mergers

Galaxy(without gas)

Galaxy(without gas)

Resulting in Boxy Ellipticals?

(frequently occurs after many previous mergers, when mass is higher)(e.g., between two elliptical galaxies)

(e.g., between two spiral galaxies)

Due to the impact of dynamical friction, halos of galaxies efficiently merge with one another to build larger systems.

Barnes & Hernquist

8. Stars dont collide calculate from typical density, velocity, and cross-section of stars. To understand equilibrium, one can ignore stellar collisions.The main challenge is to understand a system with > 1011 particles movingin their common gravitational potential.

9. Individual interactions between stars can be ignored. Hubble sequence:ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers Therelaxation time is the time needed to a↵ect the motion of the star by indi-vidual star-star interactions. This timescale is very high (1016 years) - toohigh to be relevant for galaxies. As a result, we can describe the potentialas a smooth, 3D, potential ⇢(~x)), instead of the potential produced by 1011

particles. Similarly, we can derive the phase-space density (a 6D functionf(~x,~v)), by calculating the average density in space and velocity of the 1011

or more particles.

10. Galaxy formation is driven by the formation of the halo At very highredshift, the density of the universe is close to critical. Any perturbationwill drive the density to above critical. Regions with density above criticaldensity will collapse and become self-gravitating.

11. Galaxies form inside the collapsed halos, as the gas can cool radia-tively, become self-gravitating, and form stars.

12. The equations of motion for a self-gravitating system are simple.They imply that any system can be scaled in mass, size, and velocity if thescaling factors satisfy one single relaiton This implies that any equilibriummodel can be scaled in mass and size !

13. In practice, the halos grow in mass due to merging with smaller halos.Merging is very e↵ective in the universe, due to the fact that the halos areextended and not point-sources.

14. The time-evolution of the distribution function is defined by the dis-tribution function at that time, spatial derivatives, and the gradients in thepotential (Vlasov-Equation). This follows directly from the conservation ofstars.

2

REVIEW (from Bachelor course)


Even galaxies on hyperbolic orbits, i.e., formally unbound, can merge!

What is dynamical friction?

It is drag force that collisionless gravitationally bound systems experience as they pass by each other.

8. Stars dont collide calculate from typical density, velocity, and cross-section of stars. To understand equilibrium, one can ignore stellar collisions.The main challenge is to understand a system with > 1011 particles movingin their common gravitational potential.

9. Individual interactions between stars can be ignored. Hubble sequence:ellipticals, S0s, spirals (barred and unbarred), Sa-Sd, irregulars, mergers Therelaxation time is the time needed to a↵ect the motion of the star by indi-vidual star-star interactions. This timescale is very high (1016 years) - toohigh to be relevant for galaxies. As a result, we can describe the potentialas a smooth, 3D, potential ⇢(~x)), instead of the potential produced by 1011

particles. Similarly, we can derive the phase-space density (a 6D functionf(~x,~v)), by calculating the average density in space and velocity of the 1011

or more particles.

10. Galaxy formation is driven by the formation of the halo At very highredshift, the density of the universe is close to critical. Any perturbationwill drive the density to above critical. Regions with density above criticaldensity will collapse and become self-gravitating.

11. Galaxies form inside the collapsed halos, as the gas can cool radia-tively, become self-gravitating, and form stars.

12. The equations of motion for a self-gravitating system are simple.They imply that any system can be scaled in mass, size, and velocity if thescaling factors satisfy one single relaiton This implies that any equilibriummodel can be scaled in mass and size !

13. In practice, the halos grow in mass due to merging with smaller halos.Merging is very e↵ective in the universe, due to the fact that the halos areextended and not point-sources.

14. The time-evolution of the distribution function is defined by the dis-tribution function at that time, spatial derivatives, and the gradients in thepotential (Vlasov-Equation). This follows directly from the conservation ofstars.

2

REVIEW (from Bachelor course)

What is dynamical friction?

Consider a system with high mass passing through a sea of particles with lower mass m.

The wake of lower mass particles is an overdensity and will impose a gravitational pull on high mass system, slowing it down.

The amplitude of the wake is proportional to GM, which means the gravitational force on high mass system is proportional to (GM)2

3-12-07 see http://www.strw.leidenuniv.nl/˜ franx/college/ mf-sts-07-c9-7

or

f = 5

Hence, a cooling gas cloud will settle into a disk whenit has contracted by about a factor of 5 or so.

Ellipticals and bulges can be made out of pre-existingdisks when galaxies merge, or during mergers

Homework assignment:2) Our Milky Way galaxy has a mass which is typicalfor galaxies. It likely has an isothermal dark matterhalo. Calculate the temperature of isothermal gas inthe halo of our galaxy if it has a density proportional tor−2, where r is the radius from the center.


Why do halos/ galaxies merge so easily ?

BT 7.1 abbreviated

• dynamical friction slows down two galaxies meetingeach other

• even galaxies with hyperbolic orbits can merge !

Dynamical friction

consider massive point mass M , moving in sea of smallpoint masses m• the motion of the massive object will produce a

wake behind it• the wake is an overdensity, and will exert a force

which slows down the moving massive object• the wake is proportional to GM , hence the grav-

itational force on the object is proportional toG2M2

the friction can be estimated effectively by consideringindividual interactions between the massive object,and the small point masses

The motion of the high mass system creates a wake of smaller mass particles behind it.

V0


The motion of the pointmass and the massive object issimply given by the solution for the two-body problem.

m is the mass of small pointmass.M is the mass of big pointmass.V0 is velocity difference at the start. V0 = Vm − VMb is impact parameter.The deviations in velocity are

|∆vM,⊥| =2mbV 3

0

G(M + m)2

!

1 +b2V 4

0

G2(M + m)2

"−1

|∆vM,||| =2mV0

(M + m)

!

1 +b2V 4

0

G2(M + m)2

"−1

the parallel deviation ∆vM,|| will always be in the samedirection, whereas the perpendicular component willbe in a random direction.

Hence, as a result, when the contributions from all par-ticles are taken (by integration), the perpendicularcomponent can be ignored to first order, and theparallel component will dominate.

This will slow down the moving mass M .


Assume that phase-space density of stars is f(v). Mencounters stars with velocities in volume dvm atimpact parameters between b and b + db

rate = 2πb db × V0 × f(vm)dvm

Integrate over impact parameters b, then the change invM is

dvM

dt= V0f(vm)dvm

# bmax

0

2mV0

M + m×

!

1 +b2V 4

0

G2(M + m)2

"−1

2πbdb

where bmax is the largest impact parameter that needsto be considered.Now calculate the integral, using V0 = vm − vM . Weget:

dvM

dt= 2π ln(1+Λ2)G2m(M+m)f(vm)dvm

(vm − vM )

|vm − vM |3

where

Λ ≡bmaxV 2

0

G(M + m)

Usually Λ very large. Globular cluster in Milky Way:M = 106 , V0 ≈ 100 km/s. Λ = 4.6 103, ln(1 +Λ2) ≈ 17. In the following we use ln(1 + Λ2) ≈ 2 ln Λ.We ignore the variations of Λ with the velocities vm ofthe background stars. We assume that the galaxy isisotropic.

By explicitly considering a Keplerian orbit and conserving angular momentum, one can show that the perturbations to the velocity of the high mass particle are as follows:

The impact of dynamical friction can be quantified by considering a similar two body “collision” as we considered earlier:




|∆vM,⊥| =2mbV 3

0

G(M + m)2

!

1 +b2V 4

0

G2(M + m)2

"−1

|∆vM,||| =2mV0

(M + m)

!

1 +b2V 4

0

G2(M + m)2

"−1








dvM

dt= V0f(vm)dvm

# bmax

0

2mV0

M + m×

!

1 +b2V 4

0

G2(M + m)2

"−1

2πbdb


dvM


(vm − vM )

|vm − vM |3

where

Λ ≡bmaxV 2

0

G(M + m)


high mass particle

low mass particle

V0

(M)

(m)

On average, perturbations in the perpendicular direction will cancel. However, perturbations in the horizontal direction will add.

As before, we can integrate over all impact parameters b and considering the phase-space density of small mass particles f(v) that the high mass system M is

encountering.




|∆vM,⊥| =2mbV 3

0

G(M + m)2

!

1 +b2V 4

0

G2(M + m)2

"−1

|∆vM,||| =2mV0

(M + m)

!

1 +b2V 4

0

G2(M + m)2

"−1








dvM

dt= V0f(vm)dvm

# bmax

0

2mV0

M + m×

!

1 +b2V 4

0

G2(M + m)2

"−1

2πbdb


dvM


(vm − vM )

|vm − vM |3

where

Λ ≡bmaxV 2

0

G(M + m)


Note that the impact of dynamical friction becomes more important as the mass of the systems increase

Also note that dynamical friction depends on the density of particles in the systems under collision, not the mass of the individual particles.


The resulting integral is valid for encounters with starsin volume element dvm. We have to integrate over allvm to get the total force on M . Notice that the forcedepends on vm only through the terms. Hence we haveto integrate

!

f(vm)(vm − vM )

|vm − vM |3dvm

The equation simplifies if the distribution function isisotropic. Notice that the equation is equivalent tothe equation for the gravitational force at a locationr = vM for a density denstribution ρ = f . New-tons theorem states that this is equivalent to the forcecaused by a pointmass M with mass equivalent to themass inside r.Hence

!

f(vm)(vm − vM )

|vm − vM |3dvm =

" vM

0f(vm)4πv2

mdvm

v3M

vM

Use ln(1 + Λ2) ≈ 2 ln Λ to obtain

dvM

dt= −16π2 ln(Λ)G2m(M+m)

" vM

0f(vm)v2

mdvm

v3M

vM

Only stars moving slower than vM contribute to theforce. The drag always opposes the motion.


This is the CHANDRASEKHAR DYNAMICAL FRIC-TION FORMULA (BT 7.1)

for very large vM , one derives approximately

dvM

dt= −

4π ln(Λ)G2(M + m)ρm

v2M

Two interesting points:• the drag depends on ρm, but not m itself• the larger M , the stonger the deceleration

big objects fall in faster than small objects

Dynamical friction for object with mass M inisothermal halo

For mass M in orbit in isothermal halo one derivesthe following. The density in the halo is given by ρ= σ2/(2πGr2) = v2

c/(4πGr2).Fill this in the equation above to derive:

dvM

dt= −0.428 lnΛ

GM

r2

Application to Magellanic Clouds (BT 7.11(b)

The LMC and SMC have distances of 50 and 63 kpc,and masses of 2e10 and 2e9 solar masses.

If our galaxy has a massive halo extending to the clouds,then the friction time can be estimated in the followingway:

For relative velocities between the systems vM, we can show that

where ρm is the mass density in the cloud of particles against which the high mass system is colliding.


What happens with the baryonic matter inside halos as their host halos merge? It depends.

For sufficiently low mass halos of similar masses, mergers between the baryonic matter at their centers can occur quite quickly.

However, for very high mass halos (galaxy clusters) or halos where one galaxy is substantially lower mass than the other (e.g., a dwarf galaxy merging with

the Milky Way halo), the eventual merger of two systems can take a very long time.

Why do Elliptical Galaxies Look So Similar?

Why do Elliptical Galaxies All Look So Much Alike?

i.e., why do their surface brightness profiles approximately satisfya de-Vaucouleur law?

As we have seen from the first few lectures, there is a huge freedom in constructing collisionless systems that are in

equilibrium.

So, these profiles must arise in some other way!

Violent Relaxation

One way to set up the equilibrium phase-space structure in elliptical galaxies is through violent relaxation!

Under normal conditions, for equilibrium systems, we would expect minimal changes in phase space structure of galaxies

However, what would happen if there was a collision between two galaxies?

In this case, individual stars would experience a rapidly changing gravitational potential.

How would the time-varying gravitational potential affect individual particles?Violent Relaxation II

Particle gaines energy

Particle looses energy

No Relaxation

time

Individual particles can gain or lose energy (due to the changing potential)

Credit: van den Bosch

As a result, while the total energy of the colliding galaxies does not change, there is a considerable energy exchange

between individual stars in the colliding galaxies.

The redistribution of energy between particles rapidly moves the phase space distribution to being more Maxwellian in form

(and its density profile more like an isothermal sphere)

The time scale for violent relaxation is equal to

Violent Relaxation IIIA few remarks regarding Violent Relaxation:

• During the collapse of a collisionless system the CBE is still valid, i.e., thefine-grained DF does not evolve (df/dt = 0). However, unlike for a‘steady-state’ system, ∂f/∂t = 0.

• A time-varying potential does not guarantee violent relaxation. One canconstruct oscillating models that exhibit no tendency to relax: although theenergies of the individual particles change as function of time, the relativedistribution of energies is invariant (cf. Sridhar 1989).

Although fairly artificial, this demonstrates that violent relaxation requiresboth a time-varying potential and mixing to occur simultaneous.

• The time-scale for violent relaxation is

tvr =!

(dE/dt)2

E2

"−1/2=

!(∂Φ/∂t)2

E2

"−1/2= 3

4⟨Φ2/Φ2⟩−1/2

where the last step follows from the time-dependent virial theorem (seeLynden-Bell 1967).

Note that tvr is thus of the order of the time-scale on which the potentialchanges by its own amount, which is basically the collapse time. ◃ therelaxation mechanism is very fast. Hence the name ‘violent relaxation’

where the last equation comes from the time-dependent virial theorem (Lynden-Bell 1967)

As such, the time scale for violent relaxation is basically the time scale for the merger itself -- which is a few dynamical

times. It is fast!

The redistribution of energy between particles proceeds in such a way as to be independent of the mass of stars in

galaxies (since it only occurs due to time variations in the potential).

It therefore very different from what happens during collisional relaxation where there is a dependence on the

mass of the colliding stars (e.g., as in globular clusters where the most massive stars tend to collect in the center).

Let us consider one concrete example of a system undergoing violent relaxation

One such system is a cloud of particles that start out locally at rest.

How does this system evolve with time?

Gravity will cause this cloud of particles to collapse onto the center of the cloud...

This problem was initially considered by van Albada while working in Groningen (1982)

OutlineRange of timescales

Two-body relaxation timeViolent relaxationDynamical friction

Piet van der Kruit, Kapteyn Astronomical Institute Galactic dynamics: Timescales

How does such a system evolve with time?

van Albada (1982)

Here is a short movie illustrating such a collapsing cloud:

Note that the flow is initially very uniform and ordered! After collapse, individual stars have a wide range of

velocities and energies!

How does this re-distribution of energy look for this collapsing cloud?


Furthermore, such distributions are also naturally triax-ial.

Interestingly, mergers also produce profiles which are


rather similar. This is maybe not too surprising (theprofiles are already a bit similar at the start) - and ap-parently there is enough violent relaxation during themerger to smooth things out. Here are the profilesas derived by Hopkins et al (2008) from a large set ofmerger simulations

Intrinsic correlations

We have seen that ellipticals can be scaled up anddown as far as the equilibrium laws are concerned. In

Initial Distribution

At intermediate time

At late time

(energy)

# of stars

Remarkably, it was found that the final density profile of the equilibrium system that formed from the collapsing cloud

followed a R1/4 law.

densityinitial

after collapse

R1/4 relation

What is the final density profile for this collapsing cloud?

Such density distributions that result from such a collapse are also naturally triaxial in form.








Here are projections of the density distribution of stars in the xy, xz, and yz planes.

Additional smoothing out of the phase space distribution is provided by the process of phase mixing:Phase Mixing I

Consider circular motion in a disk with Vc(R) = V0 = constant. Thefrequency of a circular orbit at radius R is then

ω = 1T

= V0

2πR

Thus, points in the disk that are initially close will separate according to

∆φ(t) = ∆!

V0

Rt"

= 2π∆ωt

We thus see that the timescale on which the points are mixed over theiraccessible volume in phase-space is of the order of

tmix ≃ 1∆ω

where due to differences in the oscillation frequencies for different stars in a collisionless system, stars spread out to

more completely fill phase space.








What about the merger between two galaxies?

Amazingly, one finds very similar density profiles to what one finds for the collapsing clouds. Here are some results

from Hopkins et al. (2008):

These merging galaxies also largely follow R1/4 density profiles.

surface brightness

What about the merger between two galaxies?

Amazingly, one finds very similar density profiles to what one finds for the collapsing clouds. Here are some results

from Hopkins et al. (2008):

These merging galaxies also largely follow R1/4 density profiles.








surface brightness

surface brightness

Intrinsic Correlations between Galaxy Properties


reality, they span a relatively narrow range in proper-ties. The figure below shows the distribution.

The figure on the top right shows the relation betweenabsolute magnitude and velocity dispersion. This isthe Faber Jackson relation, the equivalent of the Tully-Fisher relation for spirals.


The other figures show the additional relations betweensize (Re), surface brightness (< µe >), and velocitydispersion (σ). If we combine these 3 properties op-timally, we find a very tight relation (at the bottomright).

R ∝ σ1.4µ−0.9e

This relation is called the Fundamental Plane (thereis not too much fundamental about it). It is muchtighter than the Faber Jackson relation. What causessuch a relation ?

Origin of the Fundamental Plane

The fundamental plane can best be understood as arelation between the mass-to-light ratio and the massof the galaxy. Use the fundamental plane to removethe “light” term in the mass-to-light ratio:

M

L∝

M

R2eµe

∝

M

R−1.1e σ1.5R2

e

∝

M

M0.75R0.15e

∝

∝ M0.25R−0.15e

Here we used M ∝ Reσ2 . We see that the mass-to-light ratio is a function of mass (and a tiny bit ofeffective radius). This is probably due to the stellarpopulations in early-type galaxies: the metallicity, andpossibly the age of the galaxies are a function of itsmass. If so, one would automatically expect that themass-to-light ratio is a function of the mass.

Some people also think that more aspects play a rolethan just the stellar populations. It could be, for ex-ample, that the profiles of the galaxies vary systemat-ically with mass - so that the simple equations which

Strong correlations are observed between the masses, sizes, and velocity dispersions of elliptical galaxies.

average surface brightness

half-light radius

i.e., radius containing half of the

light

As we noted earlier, such correlations would not need to be present for a generic collisionless system and tell us something fundamental

about the formation of galaxies themselves.






R ∝ σ1.4µ−0.9e




M

L∝

M

R2eµe

∝

M

R−1.1e σ1.5R2

e

∝

M

M0.75R0.15e

∝

∝ M0.25R−0.15e




absolute magnitude inside the half-light radius

velocity dispersion








R ∝ σ1.4µ−0.9e




M

L∝

M

R2eµe

∝

M

R−1.1e σ1.5R2

e

∝

M

M0.75R0.15e

∝

∝ M0.25R−0.15e




velocity dispersion

average surface brightness







R ∝ σ1.4µ−0.9e




M

L∝

M

R2eµe

∝

M

R−1.1e σ1.5R2

e

∝

M

M0.75R0.15e

∝

∝ M0.25R−0.15e




half-light radius

velocity dispersion

Notice how much tighter the relationship can become

if one combines the quantities in the appropriate

fashion!


These correlations are such that galaxies populate a fundamental plane in these parameters, so that if you know two of the following three

variables for a galaxy, you can determine the third.






R ∝ σ1.4µ−0.9e




M

L∝

M

R2eµe

∝

M

R−1.1e σ1.5R2

e

∝

M

M0.75R0.15e

∝

∝ M0.25R−0.15e



where R is the size (radius), σ is the velocity dispersion, and μ is the galaxy surface brightness.

The properties of galaxies occupy a two dimensional plane in the three dimensional parameter space.

Fundamental Plane (3-D)

Here is an illustration of such a two-dimensional surface embedded in a

three-dimensional space.

How shall we interpret this fundamental plane relation?

Let us interpret in terms of the mass to light ratio of galaxies:

M

L=

M

µeRe2

µe ~ σ1.5 Re-1.1

=M

σ1.5 Re-1.1 Re2

M ~ σ2 Re

M0.75 ~ σ1.5 Re0.75

=M

M0.75 Re-0.75Re-1.1 Re2= M0.25Re-0.15

From virial relation:

The fundamental plane relation implies that the mass-to-light ratio scales as M0.25 Re-0.15.

It is unclear what causes this mass dependence. It could be due to the stellar populations (i.e., that the baryonic component in massive

galaxies have higher mass-to-light ratios) or due to dark matter playing a larger dynamical role in more massive galaxies

What about Baryonic Gas in Ellipticals? Do they have any?

Yes -- lots of hot ionized gas!

Luminosity of the hot ionized gas scales

approximately as the optical luminosity

squared!

Hot ionized gas appears to originate partially from mass

loss from AGB stars.

Heating is from SNe and movement of stars through

gas

Total gas mass ranges from 109 to 1011

solar masses

Hot Gas in Ellipticals

  LX goes as LB1.6-2.3

(empirical relation)   ρ goes as r-3/2

  Mgas~ 109–1011M

  Origin of hot gas   Reservoir   External   Internal:

  Mass return from AGB stars ~1.5 x 1011Myr-1L

-1

  Heated by Sne (what kind?), thermal motion of stars at 200-300 km s-1

Radial distribution

What about Baryonic Gas in Ellipticals? Do they have any?

Occasionally there is cool gas

However, most of the time it is outside the center of a gas.

It is expected to be rare, since elliptical galaxies do notundergo much star formation.

Dark Matter Halos

Dark Matter Halos

Two Key Steps in Galaxy Formation

1. Gravitational Collapse2. Cooling of Baryons to Center

Form of the Collapsed Dark Matter Halo(step #1 in galaxy formation process) can have an

important impact on the structure and properties of galaxies...

Let’s derive the key relations between the mass, radius, and velocity dispersion in halos

When a halo collapses -- how does ρhalo compare to ρcritical ?

Theory of Dark Matter Halos

M = (4π(r200)3/3) 200 ρcritical


The total mass of a halo is given by the following:

ρcritical = (3H(z)2)/(8πG)

The ρcritical is as follows:

As such,

M = 100 H(z)2 (rhalo)3 /G

where r200 is the radius of the halo


(V200)2 = GM/r200

From the virial relationship:

One can write expressions for the mass and radius of the haloin terms of the circular velocity and the Hubble constant:

M = (V200)3 / 10GH(z)

r200 = V200/10H(z)

The Hubble “constant” H(z) increases towards higher redshifts, effectively scaling as (1+z)3/2 at very high redshifts.

Halos of a given mass are more compact at high redshift.

Navarro-Frenk-White Density Profiles


4. Structure of Dark Matter halos

Obviously, we cannot observe the dark matter halosdirectly - but we can measure the total density profilesof clusters, and we can check in our simulations whatwe expect for dark matter halos.

4.1 Theory of halo collapse

We define a halo as a region in which the density islarger than 200 times the critical density ρcr. This def-inition serves well to describe the collapsed part of thehalo.Hence the mass M is given by

M =4π

3r3

200200ρcr

Since the critical density is given by

ρcr(z) =3H2(z)

8πG

we find

M =100r3

200H2(z)

G

Hence the halo mass and the size are uniquely related -and the relation changes as a function of redshift.The virial velocity of the halo is the circular velocity atthe virial radius

V 2

200 =GM

r200


Hence the halo mass, virial radius, and virial velocityare related by

M =V 3

200

10GH(z)

r200 =V200

10H(z)

The Hubble constant H(z) increases with lookbacktime. Hence at higher redshift, the size of a halo of agiven mass is smaller than the size of a halo with thesame mass at low redshift. Halos are more compactand denser at high redshift, which is exactly what weexpect, as the density of the universe is higher at highredshift (by (1 + z)3).Simulations show that the density profiles of halos arewell approximated by a Navarro Frenk White profile(1997):

ρ(r) =ρs

(r/rs)(1 + r/rs)2

where ρs and rs are scaling parameters. Hence at verysmall radii (r < rs), ρ ∝ r−1, and at large radii (r >rs) ρ ∝ r−3. At rs, the slope of the density profilebends over.We define a concentration parameter c by c = r200/rs.It is easy to show that

ρs =200

3ρcr(z)

c3

ln(1 + c) − c/(1 + c)

Hence the profile of the halo is completely determinedby the mass M (or equivalently, the radius r200), andthe concentration paramater c.

Simulations show that collapsed halos approximately have the following density profile:

where rs and ρs are some scaling parameters.

At small radii (r < rs), the density profile ρ scales approximately as r−1

and at large radii (r > rs), the density profile ρ scales approximately as r−3

At r ~ rs, the density profile ρ changes slope


Simulations show that the concentration index is stronglycorrelated with the mass and the redshift. Approxi-mately

c ∝

M−1/9

M∗

(1 + z)−1

where M∗ is the charateristic halo mass at a givenmass (similar to the Schechter L∗ for the luminosityfunction of galaxies, and z is the redshift of the halo.

Hence low mass halos have higher concentration.

The figures below shows the fits to halos from simula-tions


Here are some examples of the density profiles from simulations:

SCDM = standard cold dark matter

(no dark energy)

ΛCDM = cold dark matter with

dark energy

rs



c ∝

M−1/9

M∗

(1 + z)−1




13-4-12see http://www.strw.leidenuniv.nl/˜ franx/college/galaxies12 12-c04-4Here are some examples of the density profiles from simulations:

SCDM = standard cold dark matter

(no dark energy)

ΛCDM = cold dark matter with

dark energy


Define concentration parameter c = r200 / rs


4. Structure of Dark Matter halos

Obviously, we cannot observe the dark matter halosdirectly - but we can measure the total density profilesof clusters, and we can check in our simulations whatwe expect for dark matter halos.

4.1 Theory of halo collapse

We define a halo as a region in which the density islarger than 200 times the critical density ρcr. This def-inition serves well to describe the collapsed part of thehalo.Hence the mass M is given by

M =4π

3r3

200200ρcr

Since the critical density is given by

ρcr(z) =3H2(z)

8πG

we find

M =100r3

200H2(z)

G

Hence the halo mass and the size are uniquely related -and the relation changes as a function of redshift.The virial velocity of the halo is the circular velocity atthe virial radius

V 2

200 =GM

r200


Hence the halo mass, virial radius, and virial velocityare related by

M =V 3

200

10GH(z)

r200 =V200

10H(z)

The Hubble constant H(z) increases with lookbacktime. Hence at higher redshift, the size of a halo of agiven mass is smaller than the size of a halo with thesame mass at low redshift. Halos are more compactand denser at high redshift, which is exactly what weexpect, as the density of the universe is higher at highredshift (by (1 + z)3).Simulations show that the density profiles of halos arewell approximated by a Navarro Frenk White profile(1997):

ρ(r) =ρs

(r/rs)(1 + r/rs)2

where ρs and rs are scaling parameters. Hence at verysmall radii (r < rs), ρ ∝ r−1, and at large radii (r >rs) ρ ∝ r−3. At rs, the slope of the density profilebends over.We define a concentration parameter c by c = r200/rs.It is easy to show that

ρs =200

3ρcr(z)

c3

ln(1 + c) − c/(1 + c)

Hence the profile of the halo is completely determinedby the mass M (or equivalently, the radius r200), andthe concentration paramater c.

Requiring the total mass in the halo is equal to 200 ρcrit(z) (4π/3 (r200)3), one can show that

so that the density profile of a halo is completely determined by its mass and concentration parameter c.


From simulations, one finds that the concentration parameter c for a halo is closely related to its formation redshift:



c ∝

M−1/9

M∗

(1 + z)−1





Lower mass halos have higher concentration parameters.

Navarro-Frenk-White Density Profiles4 J. C. Munoz-Cuartas et al.

has a smoothing effect, not all unrelaxed haloes have a highρrms. In order to circumvent these problems, we combine thevalue of ρrms with the xoff parameter, defined as the distancebetween the most bound particle (used as the center for thedensity profile) and the center of mass of the halo, in unitsof the virial radius. This offset is a measure for the extentto which the halo is relaxed: relaxed haloes in equilibriumwill have a smooth, radially symmetric density distribution,and thus an offset that is virtually equal to zero. Unrelaxedhaloes, such as those that have only recently experienceda major merger, are likely to reveal a strongly asymmetricmass distribution, and thus a relatively large xoff . Althoughsome unrelaxed haloes may have a small xoff , the advan-tage of this parameter over, for example, the actual virialratio, 2T/V , as a function of radius (e.g. Maccio, Murante& Bonometto 2003), is that the former is trivial to evaluate.Following Maccio et al. (2007), we split our halo sample intounrelaxed and relaxed haloes. The latter are defined as thehaloes with ρrms < 0.5 and xoff < 0.07. About 70% of thehaloes in our sample qualify as relaxed haloes at z = 0.

To check for the effect of changing the definition of re-laxed haloes, we have computed the median concentration(as shown in the next section) using different values of xoff .Changing the value of this parameter by 25% (above andbelow 0.07) induces changes no larger than 5% in the me-dian concentration of dark matter haloes. We conclude thatchoosing xoff = 0.07 our results are robust enough againstvariations in the definition of relaxed population of haloes.In what follows we will just present results for haloes whichqualify as relaxed.

3 CONCENTRATION: MASS AND REDSHIFT

DEPENDENCE

In figure 1 we show the median cvir−Mvir relation for relaxedhaloes in our sample at different redshifts. Haloes have beenbinned in mass bins of 0.4 dex width, the median concentra-tion in each bin has been computed taking into account theerror associated to the concentration value (see 2.1.1, andM08). In our mass range the cvir−Mvir relation is well fittedby a single power law at almost all redshifts. Only for z = 2we see an indication that the linearity of the relation in logspace seems to break, in agreement with recent findings byKlypin et al. (2010).

The best fitting power law can be written as:

log(c) = a(z) log(Mvir/[h−1 M⊙]) + b(z) (5)

The fitting parameters a(z) and b(z) are functions ofredshift as shown in figure 2. The evolution of a and b canbe itself fitted with two simple formulas that allow to recon-struct the cvir −Mvir relation at any redshifts:

a(z) = wz −m (6)

b(z) =α

(z + γ)+

β(z + γ)2

(7)

Where the additional (constant) fitting parameters havebeen set equal to: w = 0.029, m = 0.097, α = −110.001,β = 2469.720 and γ = 16.885. Figure 3 shows the recon-struction of the cvir−Mvir relation for different mass bins asa function of redshift using the approach described above. Itshows that our (double) fitting formulas are able to recover

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 15

log 1

0(c v

ir)

log10(Mvir[M⊙ h-1])

z=0.00z=0.23z=0.38z=0.56z=0.80z=1.12z=1.59z=2.00

Figure 1. Mass and redshift dependence of the concentrationparameter. The points show the median of the concentration ascomputed from the simulations, averaged for each mass bin. Linesshow their respective linear fitting to eq. 5.

the original values of the halo concentration with a precisionof 5%, for the whole range of masses and redshifts inspected.It has been shown by Trenti et al. (2010) that using Nvir be-tween 100 and 400 particles is enough to get good estimatesfor the properties of halos, nevertheless in order to look forsystematics we re-computed cvir varying the minimum num-ber of particle inside Rvir, using 200, 500 and 1000 particles.No appreciable differences (less than 2%) were found in ourresults for the median. We also checked that our results donot change notably by changing the definition of “relaxed”haloes (i.e. changing the cut in xoff or ρrms).

It is interesting to compare our results with M08, whichshares some of the simulations presented in this work. Ourresults for the cvir −Mvir relation at z = 0 are slightly dif-ferent to those presented in M08: we found a = −0.097 andb = 2.155, while M08 found a = −0.094 and b = 2.099.The difference is less than 3%, and it is mainly due to lowmass haloes. In this work we included three new simulations,B30, B90 and B3002. Two of these (B30 and B90) increasethe statistics of our halo catalogs at the low mass end, pro-viding a better determination of the cvir −Mvir relation forM ≈ 1011h−1 M⊙. We are confident that the inclusion thosenew simulations led to an improvement over the results ofprevious works.

3.1 Understanding the concentration evolution

We want now to explore in more detail the physical mecha-nism driving the mass and redshift dependence of the con-centration parameter, this understanding could take us to abetter interpretation of the time evolution of the dark mat-ter density profile. As cvir is defined as the ratio between Rvir

and rs we will look at the time evolution of those quantitiesfor different halo masses, and try to see if we can extractsome physical insights about the evolution of cvir.

c⃝ 2010 RAS, MNRAS 000, 1–11

Munoz-Cuartas et al. 2010

Let’s also look at the rotational curves:



c ∝

M−1/9

M∗

(1 + z)−1





Does this make sense in terms of

V2 ~ GM(R)/R ?

galaxies: structure, dynamics, and...

Documents