Dynamics of galaxies and Clusters of Galaxiesaguilar/MiSitio/CF46C866-1B5... · Dynamics of galaxies and Clusters of Galaxies Luis A. Aguilar Guillermo Haro International School 2005

Dynamics of galaxies andDynamics of galaxies andClusters of GalaxiesClusters of Galaxies

Luis A. AguilarLuis A. Aguilar

Guillermo Guillermo Haro Haro International School 2005International School 2005

Instituto Instituto de de AstronomAstronomíaía, UNAM, UNAMMéxicoMéxico

Guillermo Haro 2005 Luis A. Aguilar([email protected])

2

This presentation forms part of the lectures on Galactic Dynamics given by Luis A. Aguilar,from the Instituto de Astronomía of Universidad Nacional Autónoma de México (UNAM),within the Guillermo Haro Advanced School on clusters of galaxies held at the Tonantzintlacampus of INAOE in Puebla, México, on July 2005. These lectures consist of 4 PowerPointpresentations and a Mathematica notebook. All material can be obtained from the GuillermoHaro or the author´s web sites


3

The purpose of these lectures is to introduce students to somedynamical phenomena important in the evolution of galaxies and

clusters of galaxies

Our main goal is understanding, rather than rigor

Galactic Dynamics is usually perceived as an arid discipline, full of mathand hard to apply to real problems

It is our intention to show how, learning to use some basic equations, wecan extract useful dynamical information that can be used to aid in our

understanding of galaxies and clusters of galaxies


4

These notes are divided in 4 lectures:

1. Basic Galactic Dynamics

In the first lecture, Poisson´s and Boltzmann´s equations are presented as the basic equations ofGalactic Dynamics. Then, after briefly reviewing what makes a dynamical system collisionless,one of the Jeans´ equations in spherical coordinates is introduced as a quick tool to derivedynamical properties of a model built from a density profile, without having to build a self-consistent dynamical model. The Navarro, Frenk and White profile is used as an example.

A Mathematica Notebook version of this lesson is provided as well. Students with access to thisprogram can interact with this version and change parameters in the examples provided.


5


1. Basic Galactic Dynamics2. The Orbital Structure of Spherical Potentials

In the second lecture, orbits are introduced as the basic unit to build collisionless dynamicalsystems. Their link to isolating integrals of motion is shown. The Lindblad diagram isintroduced as a tool that provides a complete catalogue of all orbits supported by a sphericalpotential. A singular, truncated model, with a flat rotation curve, is used to illustrate the use ofthe Lindblad diagram.


6


1. Basic Galactic Dynamics2. The Orbital Structure of Spherical Potentials3. When the Sky Falls: Dynamical Friction and Tidal Force

In the third lecture, we examine dynamical friction and the tidal force. These are two agentsthat play a key role when large systems accrete smaller ones during merger events. We showthat dynamical friction is a two-faced phenomenon, behaving like a usual frictional force atlow speeds, but switching to a behaviour that scales inversely with velocity at large speeds.We then examine the deformation produced by tides. We show that squeezing or stretchingalong the radial direction can occur depending on the shape of the potential producing thetides. We also examine the different effect on retrograde and prograde orbits within thesystem being truncated.


7


1. Basic Galactic Dynamics2. The Orbital Structure of Spherical Potentials3. When the Sky Falls: Dynamical Friction and Tidal Force4. When the Sky Falls: Tidal Heating and Tidal Stripping

In the last lecture, we examine tidal heating and tidal stripping. These are related to the forcesseen in the previous lecture. They are the result of ordered center of mass kinetic energybeing transferred to thermal energy within interacting systems. We describe both phenomenawith simple, rough models, that give us a sense of the expected results in each case. Finally, asimple analytical model that describes the decay of the orbit of a satellite, modelled as aPlummer model, that moves initially in a circular orbit within a galaxy with flat rotationcurve, is presented as an exercise where all concepts seen before can be applied.

Basic Galactic Dynamics

Lecture 1

Guillermo Haro International School 2005


9

Basic EquationsThe two most important equations of Galactic Dynamics arePoisson’s equation, that relates the gravitational potentialwith its source, the mass density function,

�

!2" = 4#G$

and Boltzmann’s equation, which is a transport equationthat describes the evolution of a dynamical system inphase space,

�

!f

!t+! v "

!f

!! x # $% "

!f

!! v

=!f

!t

&

'

(

) col


10

Basic EquationsThe phase space distribution function contains all thedynamical information of the system: f(x,v,t).

Its projection in configuration space gives the spatial density ofthe system:

�

!(! x ,t) = f (

! x ,! v ,t)d

3v"

The mean streaming and velocity dispersions are also obtainedfrom it:

!v (!x,t) =

!v f (!x,!v,t)d

3v! ,

" 2(!x,t) = (

!v #!v )

2f (!x,!v,t)d

3v!


11

Basic Equations

�

!f

!t+! v "

!f

!! x # $% "

!f

!! v

=!f

!t

&

'

(

) col

The left-hand side of Boltzmann’s equation describes theflow of particles in a given parcel of the system as it movesin phase-space on a timescale determined essentially by thepotential

�

tdyn ! R /vrms


12

Basic Equations

�

!f

!t+! v "

!f

!! x # $% "

!f

!! v

=!f

!t

&

'

(

) col

The right-hand side of Boltzmann’s equation is the so-called“collisional term”. Its physical meaning differs from the termin the left hand side, mainly by the timescale on which thecollisions it describes, operates.

tcol

! (R / vrms)N / log(N )


13

Matters of collisionalityThe ratio of the two timescales is a measure of the collisionality of a system:

tcol / tdyn = N / log(N )

~105102~107~1011Galaxy

~10103~103102-103Cluster of galaxies

~102~10≤103~106Globular cluster

~1≤1≤1≤ ~10Stelar group

tco l/tdyntdyn (106 years)tcol (106 years)NSystem

Notice that all parameters that characterize the size of the system in physical units are gone.The ratio only depends on N


14

Matters of collisionality

The fact that the importance of collisions decreases as N increasesis a bit counterintuitive, so it deserves a bit of thought


15


The fact that the importance of collisions decreases as N increasesis a bit counterintuituve, so it deserves a bit of thought

Let us imagine a self-gravitating system of N particles in virial equilibrium,it is clear that as time passes some collisions will occur.


16


The fact that the importance of collisions decreases as N increasesis a bit counterintuitive, so it deserves a bit of thought

Let us now imagine that we split each particle in two, thus doubling thenumber of particles in the system without perturbing the virial equilibrium.It is clear that the number of collisions per unit time will double.

So, how can we saythat the importance ofcollisions diminishes?


17


In the case of cars, collisions imply physical contact, and the faster therelative velocity, the larger the damage produced.

In the case of stars, we have long range interactions, where the largerthe relative velocity, the shorter the interaction time, and less damagewill occur.

What matters is not the number of collisions, but the effect theyhave in the system.


18

Matters of collisionalityWhat matters is not the number of collisions, but the effect in the system

In the case of our hypothetical system with N particles, the strength of the gravitationalforce goes as the product of the masses divided by their separation:

F ! (mim) / r

2

When we split each particle in two, we doubled the number of particles (and collisions), butthe mass was halved, so the numerator got reduced by a factor of 1/4:

m! m / 2 " F! F / 22

It’s true that the mean separation between particles got reduced as well, which drives theeffect in the other direction, but this effect can’t overcome the mass effect:

r ! N

"1/3# F$ F % (1 / 2

2)i(2

2/3) =F % 2

"4 /3

Even after multiplying by 2 to account for the larger number of collisions, the net effect shrinks.


19

Collisionless Galactic DynamicsSo, the more particles, the more collisions, but the less effect they willhave. When the effect of collisions becomes negligible over the timescalesof interest, we are lead to the collisionless Boltzman equation:

! f

!t+!v "

! f

!!x# $% "

! f

!!v= 0

This is a partial differential equation, that together with Poisson’s equation and the properboundary conditions, presumably set by observations, can in principle be solved to buildself-consistent dynamical models.In practice, there is not enough observational information and furthermore, the mathematicalcomplexity of the task of solving it, makes the direct solution approach an impossible taskfor realistic cases.


20

Building self-consistent modelsSeveral approaches have been used. In particular, for collisionless systems likedark halos of CDM, the so-called Jeans’ Theorem helps us to find solutions:

Any steady-state solution of the collisionless Boltzmann equationdepends on the phase-space coordinates only through integrals ofmotion in the galactic potential, and any function of the integralsyields a steady-state solution of the collisionless Boltzmann equation

�

!f

!t= 0 " f = f (Ii)

This lead to a whole industry of model building:

f = f (E), f = f (E,L), f = f (E,L2), etc.

Sir James Jeans (1877-1946)


21

Getting around building self-consistent modelsAlthough highly desirable, building a self-consistent dynamical model is notnecessary in many occasions. We could, for instance, have a density profileand just want to explore what type of kinematics this profile can support.

A very useful tool for a more direct approach that bypasses altogether the need tobuild a self-consistent model is given by Jeans’ equations, which are just thevelocity moments of the collisionless Boltzmann equation.

In the remaining of this first lecture, we will use the Jeans’ equation that results fromtaking the 2nd. velocity moment in spherical coordinates:

d

dr(!"

r

2) +

!

r[2"

r

2 # ("$

2+"%

2)] = # !

d&

dr

We will take a popular density profile in the cosmological literature,the NFW profile, and demonstrate how to extract kinematicalinformation from it without building a self-consistent model.


22

The NFW profileThe NFW profile was proposed by Navarro, Frenk & White (1997) as auniversal density profile produced by hierarchical clustering.

!(r) = !o(r / r

o)"1(1 + r / r

o)"2

It is convenient to write it in dimensionless form, bydefining a dimensionless radius:

! " r / ro

! (" ) # $(r) / $o= " %1

(1+" )%2

We notice that this profile has a cusp at the centerand goes to zero at infinity

lim!"0

# (! )"!$1 "%, lim!"%

# (! )"!$3 " 0

The NFW profile has a varying slope in the log-log plane. It diverges as 1/ζ for ζ→0 andgoes as 1/ζ3 for ζ→∞. The midpoint for this slope variation is at ζ~1.


23

The NFW profile cumulative massThe mass enclosed within a sphere of radius r, is:

Mr! 4" #(r ')r '2 dr '

0

r

$ = 4"#o(r '/ r

o)%1(1 + r '/ r

o)%2 r '2 dr '

0

r

$

= 4"#oro

3 (1+& ')%2& 'd& '0

&

$= 4"#

oro

3 [log(1+& )+ (1+& )%1 %1]

Again, it is convenient to define a dimensionlessenclosed mass:

µ(! ) " Mr/M *

= log(1+! )+ (1+! )#1 #1,

where we have defined a characteristic mass: M* ! 4"#

oro

3.

We notice the following values and limits: µ(! ) = 1 " ! = 5.3054....,

µ(! = 1) = 0.193147...,

lim!#0

µ(! ) = 0, lim!#$

µ(! ) = log(! )#$.

Notice that the massdiverges, but it does itlogarithmically.


24

The NFW profile potentialThe potential of a spherical mass distribution can be calculated as,

!(r) = " 4#G1

r$(r ')r '2 dr '

0

r

% + $(r ')r 'dr 'r

&

%'

()

*

+,

The first integral is,

!1(r) = "

4#Gr

$(r ')r '2 dr '0

r

% = "G

r$(r ') 4#r '2 dr '

0

r

% = "GM

r

r

= "GM

*

ro

µ(& )&

= " 4#G$oro

2 µ(& )&


25

The NFW profile potentialThe potential of a spherical mass distribution can be calculated as,

!(r) = " 4#G1

r$(r ')r '2 dr '

0

r

% + $(r ')r 'dr 'r

&

%'

()

*

+,

The first integral is,

!1(r) = "

4#Gr

$(r ')r '2 dr '0

r

% = "G

r$(r ') 4#r '2 dr '

0

r

% = "GM

r

r

= "GM

*

ro

µ(& )&

= " 4#G$oro

2 µ(& )&

while the second integral is,!2(r) = " 4#G $(r ')r 'dr '

r

%

& = " 4#G$oro

2 1

(1+' ')2d' '

'

%

&

= "4#G$oro

2(1+' )"1


26

The NFW profile potentialPutting everything together, !(r) = "4#G$

oro

2 % "1 log(1+% )

It is natural then to define a dimensionless potential as:where φ(r) = φoΨ(ζ), and φo ≡ -4πGρoro

2!(" ) = " #1 log(1+" )

lim!"0

#(! ) = 1,Since it is clear that φo is the depth of the potential well.

lim!"#

$(! ) = 0.We also note that

Finally, Ψ(ζ)>0 and φ(r) <0, since φo<0.

Although the mass diverges, the potentialdepth is finite, this is due to the mildlogarithmic divergence


27

The NFW profile forceWe can now compute the magnitude of the force exerted by this model:

F(r) = !d"

dr= !

"o

ro

d#

d$= !

"o

ro

F ($ )

The dimensionless force is given by,

F (! ) "d#

d!=

! $ (1+! )log(1+! )

! 2 (1+! )

The limits of the dimensionless force are:

lim!"0

F (! ) = # 12, lim

!"$F (! ) = 0

Notice that the force is discontinuous at the origin.This is due to the central cusp of the profile


28

The NFW profile: Escape andcircular velocities

So, despite the infinite mass of the model, the escape velocity is everywhere finite.

The escape velocity is easily obtained by thecondition of zero energy:E = (1 / 2)v

esc

2+ !(r) = 0 " v

esc

2= # 2!(r) = # 2!

o$(% )

It is clear that the natural unit of velocity is !o

We can then define a dimensionless escapevelocity as: !

esc

2 " vesc

2 /#o= 2 log(1+$ ) /$

The limits are: lim!"0

#esc

2= 2, lim

!"$#esc

2= 0

Escape

The circular velocity is obtained from the centrifugal equilibrium condition:

vc

2 / r = ! F(r) " #c

2= !$F ($ ) =

(1+$ )log(1+$ ) !$

$ (1+$ )

The limits in this case are, lim!"0

#c

2 $! " 0, lim!"%

#c

2= 0

Notice that the rotation curve rises as √ζ from the center, reaches a maximum of βcmax ≈0.465… at ζ

≈2.16258…, and then goes down very gently, falling 10% of its peak value at ζ≈6.66

Circular


29

The NFW profile: Escape andcircular velocities

The ratio of escape velocity to circularvelocity goes to infinity at the center,while at large radii it goes to √2, which isthe Keplerian value.

We’ll come back to this ratio later onlecture 3 when we talk about dynamicalfriction.


30

The NFW profile: velocity dispersionUp to now, all properties we have derived from the NFW profile have not required any informationwhatsoever about the velocity distribution of the model. Even the escape and circular velocities wehave derived are not diagnostics of the velocity distribution of the model, but rather characterizationsof the potential

The question thus arises as to the range of variations that are possible in thevelocity distribution as a function of position for the NFW profile.

This is an important issue, because at least in the case of elliptical galaxies, although there is somehomogeneity in the surface brightness profiles and isophotal shapes, there is a range of variation inthe velocity dispersion profiles, which could be interpreted as changes in the velocity distributions,or in the light to mass ratio. In this lecture we will explore the first possibility

One possible approach is to build a complete dynamical model and find the range of phase-spacedistributions f(x,v), which project onto the same ρ(r) when integrated over velocity space


31

The NFW profile: velocity dispersionAnother, more limited but simpler approach which is quite useful, is to useJeans’ equations to impose restrictions, not in the velocity distribution but in itsmoments, in particular in the velocity dispersion

We will assume no net rotation and a velocity distribution that is invariant under rotations.This means that the two components of the tangential velocity dispersion are equal: σθ

2= σϕ2,

and the velocity ellipsoid everywhere can be characterized by its radial velocity dispersion σr2

and an anisotropy parameter:! " 1#

$%2

$r

2

Notice that β is negative for a model dominated by tangential motions, goes to 0 for theisotropic case, and reaches 1 for the purely radial motion case

β

10

vr

vθ

vϕ


32

The NFW profile: velocity dispersionAnother, more limited but simpler approach which is quite useful, is to useJeans’ equations to impose restrictions, not in the velocity distribution but in itsmoments, in particular in the velocity dispersion

We will assume no net rotation and a velocity distribution that is invariant under rotations.This means that the two components of the tangential velocity dispersion are equal: σθ

2= σϕ2,

and the velocity ellipsoid everywhere can be characterized by its radial velocity dispersion σr2

and an anisotropy parameter:! " 1#

$%2

$r

2

Notice that β is negative for a model dominated by tangential motions, goes to 0 for theisotropic case, and reaches 1 for the purely radial motion case

In spherical coordinates, the Jeans’ equation that corresponds to the 2nd.velocity moment of Boltzmann’s equation is:

d

dr(!"

r

2) +

!

r[2"

r

2 # ("$

2+"%

2)] = # !

d&

dr


33

The NFW profile: velocity dispersionThe isotropic case

The first case that we will study is the model whose velocity distribution is isotropic everywhere.From the Jeans’ equation on last slide we obtain (β =0):

1

!d

dr!" 2( ) = #

d$dr

% " 2(r) =

1

!(r)!(r ')

d$dr '

dr '

r

&

'

In dimensionless form, this equation is: ! 2(" ) = #

1

$ (" )$ (" ')

d%d" '

d" '"

&

'! " # / $

oWhere we have defined the dimensionless velocity dispersion as:

Using our previously defined dimensionless density and force functions, we can evaluate thisprevious expression:

! 2 (" ) = #" (1+" )21

" '(1+" ')2" '# (1+" ') log(1+" ')

" '2 (1+" ')d" '

"

$

%

= #" (1+" )2" '# (1+" ') log(1+" ')

" '3(1+" ')3d" '

"

$

%The integrand is a function everywhere positive that diverges as ζ-1 at the origin and approaches0 at large radii. We can not integrate it analytically, but we can do it numerically.


34

The NFW profile: velocity dispersion The isotropic case

The velocity dispersion of the isotropic model is quiteflat, it goes to zero at the origin and at large radii,reaching a maximum of χ ≈ 0.30707… at ζ ≈ 0.7625…

The shrinking velocity dispersion at the center is a result ofthe mild divergence of the density cusp, it diverges as ρ∝ζ-1.


35

!(r)" 2(r) = !(r ')

d#dr '

dr '

r

$

% ,




We can see this as follows: the equation we used toobtain σ(r) can be written as,


36




We can see this as follows: the equation we used toobtain σ(r) can be written as,

!(r)" 2(r) = !(r ')

d#dr '

dr '

r

$

% ,

the left-hand side is the local amount of kinetic energy per unit volume, or local pressure. The right-hand side isthe force per unit volume integrated on a radial column from the local position all the way to infinity; this is theforce per unit area that the local element has to support.


37


!(r)" 2(r) = !(r ')F(r ')dr '

r

#

$ ,

If we assume that ρ ∝ rα, it is clear that as r→0,

! > " 2 # $2% 0,

! = " 2 # $2% constant,

! < " 2 # $2%&,

so a cusp that is steeper than r-2 is required to force a divergent central isotropic velocity dispersion

F(r)!Mr/ r

2 ! "r3 / r2 ! " r # "Fdr$ ! "2r2 ,

so, the local pressure ρσ2 has to go as ρ2r2, or σ 2∝ρr2.

Now, in a spherical mass distribution, the force goes as theenclosed mass divided by the radius squared and the enclosedmass goes as the density times the radius cubed:

Another way of looking at this, is that you need to pack a lot of mass at the center, so thatthe resulting gravitational force makes the local velocity dispersion to soar without bound.


38

The NFW profile: velocity dispersion The radial case

As an extreme case, we will now explore the possibility of building an NFW model where all the orbitsare radial. This would maximize the observed central velocity dispersion.

Our next step would be to write the dimensionless form of this equation and substitute the appropriatefunctions for the density and the potential of the NFW profile.This can be done, and indeed, it gives an answer that diverges strongly at the center. However, it isvery important to emphasize that, although the Jeans’ equation can be formally solved, the impliedsolution may not be physical

We begin by multiplying both sides of the general equation by (ρr2):

our next step is to realize that,

which is easily integrated to yield (β=1),

the first equation can then be written as,

r2 d

dr!"

r

2( ) + 2#r !"r

2( ) = $ !r2d%

dr,

d

drr2!"

r

2( ) = r2d

dr!"

r

2( ) + 2r !"r

2( ),

d

drr2!"

r

2( ) # 2r !"r

2( ) 1# $( ) = # ! r2 %d

drr2!"

r

2( ) = ! r22"

r

2

r(1# $)#

d&dr

'

()

*

+,,

!r

2(r) =

1

r2"(r)

r '2 "(r ')

d#dr '

dr '

r

$

%

In principle, one can solve the relevant Jean’s equation for the general case β ≠0 as follows:


39


In the particular case of purely radial orbits, we should realize that we are putting a very strongconstraint on the central density: since all orbits are radial, all go through the center, and so the centralregion must accommodate all particles, although not at the same time.

It can be proved that the solution obtained from the Jeans’ equation for the purely radial orbits in theNFW case, implies a phase-space distribution function that becomes negative at the center, somethingthat is clearly non-sensical.This is the mathematical way of the formal solution to accommodate all particles on radial orbitswithin a central cusp that does not diverge quickly enough.

We will examine in more detail this question and derive a general lower limitto the rate of divergence that a central cusp must have to accommodate apopulation of particles in radial orbits.


40


The relation between the phase-space distribution function and the spatial density is: !(r) = f (r,v)d3v"

In a system that is integrable, the f(r,v) must be expressable as a function of the integrals of motion (Jeans’ Theorem).Now, if the system is invariant with respect to spatial rotations, as we are assuming here, then we can take f=f(E,L2),since the energy E and the magnitude of the angular momentum L are invariant with respect to rotations.We can then write the density integral as,

!(r) = f (E,L2)d

3v" = 2# dvr

$%

%

" vt dvt0

%

" f (12vr2+ 1

2vr2+ &, r2vt

2),

where we have separated the integration over velocity space in two parts, one over the radial direction and theother over the tangential plane.

Now, because we are building a model with radial orbits only, the phase-space distribution can be written as:

f (E,L2) = g(E)! (L2 ) = g(12vr

2+ ")! (r2vt

2),

where the radial velocity dependence is in the g function and the tangential velocity dependence is in the δfunction, which is a Dirac delta function:

! (x) = 0 " x # 0, ! (x)dx$%

%

& = 1.


41


The density integral can then be split in two factors:

!(r) = 2" g(12vr2+#)dvr

$%

+%

& i ' (r2vt2)vtdvt

0

%

&

To do the integral on the tangential velocity we change the integration variable: x = r2vt

2! dx = 2r

2vtdv

t,

! (r2vt

2)v

tdv

t

0

"

# = ! (x)vt

dx

2r2vt0

"

# =1

2r2

! (x)dx0

"

# =1

2r2

Putting this result back in the density integral,

!(r) = 2" g(12vr2+ #)dvr

$%

+%

& i1

2r2

' r2!(r) = " g(12vr

2+ #)dvr

$%

+%

&Let’s assume that at the center, the density profile behaves as ρ ∝ rα, then it is clear that r2ρ(r)→0 for α>-2,forcing the left hand side of our result to go to 0 at the origin. However, the right hand side is an integral overg(E), which is a positive function, and so the only way this integral can be 0 is if g(E)=0, which gives no model!

So, we conclude that the only way to build a dynamical model with radial orbits only, is tohave a central density cusp that diverges faster than 1/r2 at the center.

The NFW profile does not satisfy this condition and so no radial orbit model is possible.


42

The NFW profile: velocity dispersion The tangential case

The opposite extreme to a radial orbit model is one with tangential motion only.In a spherical model, this means that all orbits are circular and so no radialmixing exists.Such a model is always possible since we are free toput as many stars as are required by the density profileat each radius.It is easy to see that, in this case, the phase-spacedistribution function is:

f (r,vr ,vt ) =1

!"(r)# (vr )# (vt

2 $ vc2),

where vc is the local circular velocity and the tangential velocity dispersion is simply: σt = vc ,which we have already computed.

A word of caution is appropriate here, just because we can find a solution, this does notimply that it is stable. This is particularly critical for models built with circular orbits only.


43

The NFW profile Summary of kinematics

Before addressing the problem of projecting on the plane of the sky all ourkinematical results, we summarize what we have found so far

We will come back to this diagramin lecture 3, when we talk aboutdynamical friction. Isotropic

Circular

Escape

Although at the center, both theisotropic and the purely tangentialvelocity dispersions vanish, withinς<~0.2 the former is larger. Atlarger radii the purely tangentialdispersion becomes larger.


44

The NFW profile:Projected density

!(R) = 2"(r)r

r2 # R2

dr,

R

$

%

Having computed the velocity dispersion for the isotropic andtangential versions of the NFW profile, we can now compute theline of sight velocity dispersion. Our first step is to obtain theprojected density as an integral over the line of sight:

Rr

where R is the projected distanceto the center of the model

This expression is easy to obtain as an integral over the line of sight of the spatial density. If we call z thedistance along the line of sight, simple geometry leads to:

Rr

z

z2= r

2 ! R2 " dz = rdr / z = rdr / r2 ! R2

#(R) = $(r)dz!%

+%

& = 2 $(r)rdr / r2 ! R2

0

%

&


45


!(R) = 2"(r)r

r2 # R2

dr,

R

$

%

ηζIf we define a dimensionless

projected radius as η ≡ R/ro, anduse our previously defineddimensionless functions, we cancast the previous equation indimensionless form:

!(R) = 2"o# ($ ) r

o$

ro$ 2 %&2

rod$ = 2"

oro

# ($ )$

$ 2 %&2d$

&

'

(&

'

(

We can now define a dimensionless projected density as: !(") # $(R) / (2%oro)


46


!(") =d#

(1+# )2 # 2 $"2=

"2 $1 $ arcsec(")("2 $1)3/2"

%

&In the case of the NFW profile we get:

Caution should be used when evaluating this expression, as both the numerator and denominator give complexnumbers for η <1, but the ratio is always real for η >0.

lim!"0

#(!) = $, lim!"$

#(!) = 0

The projected density has the following limits:


47

The NFW profile:Projected velocity dispersion profiles

We can now compute the projected velocity dispersion for the isotropic andpurely tangential cases. This can be done using the following expression:

Rr

σr

σlos

σt

! p

2(R) =

2

"(R)#(r)! los

2(r,R)r

r2 $ R2

dr ,R

%

&

where σlos(r,R) is the line of sight velocitydispersion on a volume element atdistance r from the center and along theline of sight at projected distance R.

Elementary geometry shows that the line of sight velocity dispersion can be written as,!los

2(r,R) = (1" R

2/ r

2)!

r

2(r) + (R

2/ 2r

2)!

t

2(r) ,

where σr(r) and σt(r) are the radial and tangential velocity dispersions at r


48


In the isotropic case: !t

2= 2!

r

2" !

los

2(r,R) = !

r

2(r)

The projected velocity dispersion in theisotropic case is then,

! p" iso2

(R) =2

#(R)$(r)! r

2(r)r

r2 " R2

drR

%

& ,

or in adimensional form,

! p" iso2

(#) =1

$(#)% (& )!r

2(& )&

& 2 "#2d&

#

'

(

The corresponding expressions for thetangential case are,

! p" tan2

(R) =2

#(R)$(r)(R2 / 2r2 )! t

2(r)r

r2 " R2

drR

%

& ,

! p" tan2

(#) =#2

2$(#)% (& )!t

2(& )

& & 2 "#2d&

#

'

(


49


Using the forms previously found for χiso and χt we can evaluate these expressions and plot them

! p" iso2

(#) =1

$(#)% (& )!r

2(& )&

& 2 "#2d&

#

'

( ! p" tan2

(#) =#2

2$(#)% (& )!t

2(& )

& & 2 "#2d&

#

'

(

In the both cases the projected velocitydispersion rises from the center to amaximum at η≈0.6 (isotropic case) or η≈2.5(tangential case) and then decreases steadilyfor larger distances.

Tangential

Isotropic


50

The NFW profile:Spatial and projected kinematics

Tangential

Isotropic

Tangential

Isotropic

Comparing 3D with projected kinematics, wesee that both have a similar behaviour. However,the curves are switched: whereas the tangentialmodel dominates in 3D, the isotropic onedominates in the projected kinematics.This is due to the projection.


51

Other interesting profiles

There are two very useful, but still simple profiles, that have been used in the literatureto model spherical systems. These are the Jaffe (Jaffe 1983, Merritt 1985) and theHernquist (Hernquist 1990, Baes and Dejonghe 2002) profiles:

!Jaffe(r) = (M / 4"ro3) (r / ro )

#2(1 + r / ro )

#2,

!Hernquist (r) = (M / 2"ro3) (r / ro )

#1(1 + r / ro )

#3,

where M is the total mass of the system and ro is a scale-length whose physicalmeaning is different in each case.


52

Other interesting profiles!Jaffe(r) = (M / 4"ro

3) (r / ro )

#2(1 + r / ro )

#2,

!Hernquist (r) = (M / 2"ro3) (r / ro )

#1(1 + r / ro )

#3.

Notice that these two profiles behave as ρ ∝ r -4 at large radii, as opposed to theshallower NFW profile. This behaviour has been found in simulations of galaxyinteractions when they are isolated and not part of a cosmological expansion (e.g.Aguilar & White, 1986), and has also been observed in some real ellipticalgalaxies (Kormendy, 1977).

What is the reason for a ρ ∝ r -4 profile at large radii?


53

ρ∝r-4 profiles

An isolated, self-gravitating system has a particle distribution in energy that goesto zero at E≤0, since all particles with positive energy are not bound.

0

E

N(E)


54

ρ∝r-4 profiles

An isolated, self-gravitating system has a particle distribution in energy that goesto zero at E≤0, since all particles with positive energy are not bound.

0

E

N(E)

However, if the system is perturbed by an external agent, the particle distributionwill be spilled over the E=0 boundary, since the perturbation doesn’t care aboutthe escape velocity of the initial system.

We will show that this is enough to guarantee an ρ ∝ r -4 profile at large radialdistances


55

ρ∝r-4 profilesIf a spherical system with a 1/r potential and no rotation has an isotropic velocitydistribution that is considered to have a finite, non-zero population of particles atE=0, the local tail of the density profile will then exhibit a ρ ∝ r -4 behaviour.

For all finite mass models, at sufficiently large radial distances, thepotential function looks like that of a point mass, i.e. φ∝1/r, and sothe result implies that the density profile will behave as ρ∝1/r4.

0

E

N(E)

N(E=0)≠0

Notice that this result does not apply to the NFW profile, whose mass diverges


56

ρ∝r-4 profilesProof: E =

1

2v2+ !(r), ! < 0 " v = 2(E #!(r)

Now, the number of particles with energies between E and E+ΔE, is:

N(E)!E = f (r,v) d3r d

3v = 16" 2

#E

$ f (E)r2dr v

2dv

#E

$

where f(E) is the phase-space density and ΩE is the volumein phase space with energy between E and E+ΔE.

Since f is a function of E it is convenient to changethe velocity integral for one over energy:

N(E)!E = 16" 2f (E)r

2v2 #(r,v)

#(r,E)dr dE

$E

%

Since the Jacobian is equal to 1/v and f(E) is constantwhen integrating over E:N(E)!E = 16" 2

f (E) r2v dr dE

#E

$ = f (E)A(E)dE

ΩE

E

E+ΔE


57

ρ∝r-4 profilesProof: So, we have that:

A(E) = 16! 2r22[E "#(r)] dr

# (r )<E$ ,

where A(E) is the “area” of the constant energy surface in phase-space.

N(E)!E = f (E)A(E)dE ,


58


A(E) = 16! 2r22[E "#(r)] dr

# (r )<E$ ,


Now, if φ ∝1/r,


! A(E) " r2r#1/2

r = r5 /2

" E#5 /2


59


A(E) = 16! 2r22[E "#(r)] dr

# (r )<E$ ,



Now, if φ ∝1/r, ! A(E) " r2r#1/2

r = r5 /2

" E#5 /2

Then, if N(E ~ 0)dE = f (E ~ 0)A(E ~ 0)dE ! 0,

! f (E ~ 0) " E+5 /2

non-zero and finite,

!(r) = 4" f (E) 2[E #$(r)]dE$ (r )

0

%Now, since the density can be written as:

! " # E+5 /2

E1/2E = E

4 # r$4✔


60

For those interested in digging up more

• Fully self-consistent anisotropic models in phase space have been obtained by Baesand Dejonghe (2002) for the Hernquist profile and by Merritt (1985) for the Jaffeprofile.

• Dehnen (1993) has presented a very general class of models that includes the Jaffeand Hernquist models as special cases. Most of these models can be expressed ascombinations of elementary functions.

•Lokas & Mamon (2001) have derived many properties of spherical models with NFWprofiles, including phase space distribution functions with various kinematics.

•Widrow & Dubinski (2005) have introduced a class of self-consistent models for diskgalaxies that consist of an NFW dark halo, an exponential disk, a Hernquist bulge andeven a massive central blach hole.


61

Proposed Problems

1. Prove that the enclosed mass, potential and force functions are, in each case,

where ζ=r/ro

Mr: M! (1+! )"1 M! 2 (1+! )"2

#(r) : GM / ro( ) log[! / (1+! )] " GM / r

o( )(1+! )"1

F(r) : " GM / ro

2( )! "1 (1+! )"1 " GM / ro

2( )(1+! )"2

Jaffe Hernquist


62

Proposed Problems2. Find expressions for the escape and circular velocities for these profiles

3. Use Jeans’ equations for spherical systems with isotropic velocity distributions tofind the 3D and projected velocity dispersions for each profile. For the Jaffe profile,Jaffe (1983) is a good guide, his equation (8) is the Jeans’ equations we have beenusing. For the Hernquist profile, Hernquist (1990) is a good reference. His equation (9)is again, the Jeans’ equation we have been using. In both cases the quoted referencesonly list the final results. The student should reproduce all intermediate steps followingthe present notes as a guide and verify the results against the quoted references

4. The Jaffe profile has a central cusp that allows a purely radial orbit model. Jaffe(1983) can be used as a guide to obtain the projected velocity dispersion for this case.


63

Bibliography Aguilar, L.A. & White, S.D.M. (1986) ApJ 307, 97-109: Set of N-body simulations of interacting galaxies that results in ρ ∝ r-4 asymptotic behaviour at large radial distances.Baes, M. & Dejonghe, H. (2002) AA 393, 485-497: Anisotropic, self-consistent Hernquist models. Binney, J. & Tremaine, S. (1987) Galactic Dynamics, Princeton University Press: The standard graduate level reference in Galactic Dynamics. Section 4.2 discusses Jeans’ equations. Bouvier, P. & Janin, G. (1968) Publ. Obs. Genève, 74, 186: Earlier proof that a profile as steep as ρ ∝ r-2

is needed for purely radial orbits. Dehnen, W. (1993) MNRAS 255, 250-256: A very useful general class of models that includes the Jaffe and Hernquist models as particular cases. It also provides a completely analytical model very close to a deVaucouleurs profile. Hernquist, L. (1990) ApJ 356, 359-364: Original Hernquist model. Jaffe, W., (1983) MNRAS 202, 995-999: Original Jaffe model. Kormendy, J. (1977) ApJ 218, 333-346: Observations of elliptical galaxies with evidence of perturbation in the outer parts of their density profiles. Lokas, E. & Mamon, G. (2001) MNRAS 321, 155-166: Many properties of NFW models are derived, including phase-space distributions with various kinematical properties. Merritt, D. (1985) MNRAS 214, 25p-28p: Anisotropic, self-consistent models. Navarro, J., Frenk, C.S. & White, S.D.M. (1997) ApJ 490, 493-508: Original NFW profile. Richstone, D. & Tremaine, S. (1984) ApJ 286, 27-37: Proof of the ρ ∝ r-2 central cusp for systems with purely radial orbits. Widrow, L.M. & Dubinski, J. (2005) ApJ 631, 838-855: Presents a set of disk galaxy models with NFW halo, exponential disk, Hernquist bulge and even a central massive black hole. Good to generate N-body initial conditions.

Documents

Dynamics of galaxies and Clusters of Galaxiesaguilar/MiSitio/CF46C866-1B5... · Dynamics of galaxies and Clusters of Galaxies Luis A. Aguilar Guillermo Haro International School 2005