11th Lec

• Phase Space

Collisionless Systems

• We showed collisions or deflections are rare• Collisionless: stellar motions under influence of

mean gravitational potential!• Rational:• Gravity is a long-distance force, decreases as r-2

– as opposed to the statistical mechanics of molecules in a box

Collisionless Systems

• stars move under influence of a smooth gravitational potential– determined by overall structure of system

• Statistical treatment of motions– collisionless Boltzman equation– Jeans equations

• provide link between theoretical models (potentials) and observable quantities.

• instead of following individual orbits

• study motions as a function of position in system

• Use CBE, Jeans eqs. to determine mass distributions and total masses

Fluid approach:Phase Space Density

PHASE SPACE DENSITY:Number of stars per unit volume per unit velocity volume f(x,v) (all called Distribution Function DF).

The total number of particles per unit volume is given by:

313 )(kmspc

mN

volumevelocityvolumespace

mstarsofnumberv)f(x,

• E.g., air particles with Gaussian velocity (rms velocity = σ in x,y,z directions):

• The distribution function f(x,v) is defined by:

mdN=f(x,v)d3xd3v

where dN is the number of particles per unit volume with a given range of velocities.

3

2

222

o

)2(

2expnm

v)f(x,

xyx vvv

• The total mass is then given by the integral of the mass distribution function over space and velocity volume:

• Note:in spherical symmetry d3x=4πr2dr,

• for isotropic systems d3v=4πv2dv

• The total momentum is given by:

xdvdvxfxdxM total

333 ),()(

vdxdvvxfmdNvPtotal

33),(

• Example:mean speed of air molecules in a box of dx3 :

These are gamma functions

3

2

222

o

)2(

2expnm

v)f(x,

xyx vvv

• Gamma Functions:

2

1

)1()1()(

)( 1

0

nnn

dxxen nx

How to calculate dx3 and dv3

22 2 2

0 0

3 2 2

3 2 2

2 2 2

sin 2 2 ,

4 (if spherical)

4 (if isotropic)

2[ ( )]

x y z

x y z

d d d r x x y z

d dxdydz r drdΩ πr dr

d dv dv dv v dvdΩ πv dv

V v v v v E x

x

v

DF and its moments

mass-weighted average,

,

AdMA A

dM

3 3 3

3 3 3

2 2 2

( ) ( , )

( , )

1: A( , ) Vx , ( ) ,

2

d A dM A d Af x v d

d dM d f x v d

For Vx Vy Vz x

x x x v

x x v

x v ,VxVy x v

Additive: subcomponents add upto the total gravitational mass

1 2

1 2

A B A B

f f f

Full Notes online

• http://www-star.st-and.ac.uk/~hz4/gravdyn/GraviDynFinal3.ppt GraviDynFinal3.pdf

Liouvilles Theorem

We previously introduced the concept of phase space density. The concept of phase space density is useful because it has the nice property that it is incompessible for collisionless systems.

A COLLISIONLESS SYSTEM is one where there are no collisions. All the constituent particles move under the influence of the mean potential generated by all the other particles.

INCOMPRESSIBLE means that the phase-space density doesn’t change with time.

Consider Nstar identical particles moving in a small bundle through spacetime on neighbouring paths. If you measure the bundles volume in phase space (Vol=Δx Δ p) as a function of a parameter λ (e.g., time t) along the central path of the bundle. It can be shown that:

It can be seen that the region of phase space occupied by the particle deforms but maintains its area. The same is true for y-py and z-pz. This is equivalent to saying that the phase space density f=Nstars/Vol is constant. df/dt=0!df/dt=0!

THEOREM'LIOUVILLES',0,0 d

dNstar

d

dVol

px

x

px

x

motions in phase-space

• Flow of points in phase space corresponding to stars moving along their orbits.

• phase space coords:

• and the velocity of the flow is then:– where wdot is the 6-D vector related to w as

the 3-D velocity vector v relates to x

),(),(

),...,,(),( 621

vvxw

wwwwvx

• stars are conserved in this flow, with no encounters, stars do not jump from one point to another in phase space.

• they drift slowly through phase space

• In the COMBINED potential of stars and dark matter

fluid analogy

• regard stars as making up a fluid in phase space with a phase space density

•

• assume that f is a smooth function, continuous and differentiable– good for N >105

f (x,v,t) f (w,t)

• as in a fluid, we have a continuity equation• fluid in box of volume V, density, and

velocity v, the change in mass is then:

– Used the divergence theorem

0

3

23

23

V

SV

SV

xdvt

SdFxdF

SdvxdtdtdM

continuity equation• must hold for any volume V, hence:

• in same manner, density of stars in phase space obeys a continuity equation:

If we integrate over a volume of phase space V, then 1st term is the rate of change of the stars in V, while 2nd term is the rate of outflow/inflow of stars from/into V.

0

0

0

3

1

3

1

6

1

6

1

i iii i

i

i

i

xvv

v

x

v

w

w

w

wf

t

f

vt

0

Collisionless Boltzmann Equation

• Hence, we can simplify the continuity equation to the CBE:

• Vector form 0

0

0

3

1

6

1

v

ffv

t

f

v

f

xx

fv

t

f

w

fw

t

f

i iiii

• in the event of stellar encounters, no longer collisionless

• require additional terms to rhs of equation

CBE cont.

• can define a Lagrangian derivative• Lagrangian flows are where the coordinates travel along with the motions

(flow)– hence x= x0 = constant for a given star

• then we have:

• and

• rate of change of phase space density seen by observer travelling with star• the flow of stellar phase points through phase space is incompressible• f around the phase point of a given star remains the same0

6

1

6

w

fw

t

f

dt

df

wtdt

d

incompressible flow

• example of incompressible flow• idealised marathon race: each runner runs at

constant speed• At start: the number density of runners is

large, but they travel at wide variety of speeds• At finish: the number density is low, but at

any given time the runners going past have nearly the same speed

DF & Integrals of motion

• If some quantity I(x,v) is conserved i.e.

• Assume f(x,v) depends on (x,v) through the function I(x,v), so f=f(I(x,v)).

• Such phase space density is incompressible, i.e

0),(

dt

vxdI

0dt

df

Jeans theorem

• For most stellar systems the DF depends on (x,v) through generally three integrals of motion (conserved quantities), Ii(x,v),i=1..3 f(x,v) = f(I1(x,v), I2(x,v), I3(x,v))

• E.g., in Spherical Equilibrium, f is a function of energy E(x,v) and ang. mom. vector L(x,v).’s amplitude and z-component

)ˆ||,||,(),( zLLEfvxf

3D Analogy of 6D Phase space

• If DF(x,v) is analogous to density(x,y,z), • Then DF(E,L,Lz) is ~ density(r,theta,phi), • Integrals analogous to spherical coordinates

– E(x,v) analogous to r(x,y,z)

• Isotropic DF(E) ~ spherical density(r)– Normalization dM=f(E)dx3dv3 ~ dM=density(r)dr3

– Have non-self-gravitating subcomponents: DF1+DF2, like rho1+rho2 to make up total gravity.

12th Lec

• Phase Space

Tensor Virial Theorem• Equation of motion:

dt

vd

0 0

1 1.

T Tdvdt r dtr

T dt T

dt

rdv

dt

vrd

)(

TTT

rdtT

dtvvTT

rv

000

.11)(

rvv

This is Tensor Virial Theorem

• E.g.

• So the time averaged value of v2 is equal to the time averaged value of the circular velocity squared.

22

2

2222

)(

.

etc

cir

cir

zyx

yyx

xxx

vv

sphericalvdr

dr

r

vvvv

xvv

xvv

Scalar Virial Theorem

• the kinetic energy of a system with mass M is just

where <v2> is the mean-squared speed of the system’s stars.

• Hence the virial theorem states that 2

2

1vMK

.2

2

rv

r

GM

M

Wv

g

Virial02 WK

Stress Tensor • describes a pressure which is anisotropic

– not the same in all directions

• and we can refer to a “pressure supported” system

• the tensor is symmetric.

• can chose a set of orthogonal axes such that the tensor is diagonal

• Velocity ellipsoid with semi-major axes given by

Pnx ij

i

2

ij2 ii

2 ij

11 , 22 , 33

n ij2

Subcomponents in Spherical Equilibrium Potential

• Described by spherical potential φ(r)• SPHERICAL subcomponent density ρ(r) depends on

modulus of r.

• EQUILIBRIUM:Properties do not evolve with time.

,

,0,0 (0, ,0) (0,0, )

0 0

r r

r r r

x xy

000

ttt

f

• In a spherical potential

rrxy

dr

rd

r

yx

r

r

y

rx

y

rx

)(

)()(

r

y

dy

dr

ydyrdr

ydrd

zyxr

)()( 22

2222

So <xy>=0 since the average value of xy will be zero.

<vxvy>=0

Spherical Isotropic f(E) Equilibrium Systems

• ISOTROPIC:The distribution function f(E) only depends on the modulus of the velocity rather than the direction.

2

2 2 2 2 2 2tangential

, / 2 ( )

1

20

x y z r

x y

f E E v r

v v

Note:the tangential direction has and components

Anisotropic DF f(E,L) in spherical potential.

• Energy E is conserved as:

• Angular Momentum Vector L is conserved as:

• DF depends on Velocity Direction through L=r X v• Hence anisotropic

0t

0

2 2tangential

1

2r

e.g., f(E,L) is an incompressible fluid

• The total energy of an orbit is given by:

),(2

1 2 trvE

( , ) ( , ) ( , )0 0

df E L f E L dE f E L dL

dt E dt L dt

0 for static potential, 0 for spherical potential

So f(E,L) constant along orbit or flow

drdrdrd trr //)2(/)( 222

• spherical Jeans eq. of a tracer density rho(r)

•

• Proof :

)/()(),(//

/)(),(

/)()(

///

/22

23232

223

2223

32

222

drdvdELdLEffdvrrLdrd

rvdELdLEffdv

rvdELddvvddv

rLdrddrdvv

rLEv

r

r

rrt

rr

r

Jeans eq. Proof cont.

222

422

2

222

00

222

4222

223

/2/)(

//),(

/)/(),(

/)/(*),(/)(

),(

)/(*),(

)/(),(

2

trr

r

r

rr

r

vL

r

rt

r

drrddrrd

vrLdELdLEf

vrdrddELdLEf

rdrdvdELdLEfrdrrd

vdELEfLdr

vrLdELdLEf

rvdELdLEffdv

r

• SELF GRAVITATING:The masses are kept together by their mutual gravity.

• In non-self gravitating systems the density that creates the potential is not equal to the density of stars. e.g a black hole with stars orbiting about it is NOT self gravitating.

13th Lec

• Phase Space

Velocity dispersions of a subcomponent

in spherical potential• For a spherically symmetric system we have

• a non-rotating galaxy has – and the velocity ellipsoids are spheroids with

their symmetry axes pointing towards the galactic centre

• Define anisotropy

22

2222

/1

2

r

rr

vvr

dr

dnvvv

r

nvn

dr

d

v2 v

2

Spherical mass profile from velocity dispersions.

• Get M(r) or Vcir from:

• RHS: observations of dispersion and as a function of radius r for a stellar population.

2ln

ln

ln

ln

21

222

circ

2

22

rd

vd

rd

ndv

r

rGM

dr

drv

r

rGM

dr

d

r

vvn

dr

d

n

rr

rr

• Isotropic Spherical system, β=0

• This is the isotropic JEANS EQUATION, relating the pressure gradient to the gravitational force.

dr

dr

dr

d )(

)( 2

Note: 2=P

2 | |r r

dP dr g dr

dr

Above Solution to Isotropic Jeans Eq:

negative sign has gone since we reversed the limits.

Hydrostatic equilibrium Isotropic spherical Jeans equation

• Conservation of momentum gives:

dr

dr

dr

d

dr

dP )(

)( 2

0

1

P g

g P

Tutorial

g

(r) (r)

2

(E)vesc

M

Tutorial Question 3• Question: Show dispersion sigma is constant in potential Phi=V0

2ln(r). What might be the reason that this model is called Singular Isothermal Sphere?

24

24

0,rrAt

1

4

1

2

2

2

2

222

2o

2

2

222

2

2

cc

cc

cr

cr

c

oc

r

r

v

Gr

v

r

v

G

v

P

drr

v

rG

vdr

r

v

rrr

v

dr

d

• Since the circular velocity is independent of radius then so is the velocity dispersionIsothermal.

2

2

22

2

22

c

c

c

v

v

v

Flattened Disks

• Here the potential is of the form (R,z).• No longer spherically symmetric.• Now it is Axisymmetric

zg

Rg

zRR

RR

GzRzR

zr

2

2

4

1),(),(

Question 4: Oblate Log. potential

• oblate galaxy with Vcirc ~ V0 =100km/s

• Draw contours of the corresponding Self-gravitating Density to show it is unphysical.

• Let Lz=1kpc*V0 , E=0.55*V02 +C0, Plot

effective potential contours in RZ plane to show it is an epicycle orbit.

• Taylor expand the potential near (R,z)=(1,0) to find epicycle frequencies and the approximate z-height and peri-apo range.

2 2 210 02( , ) ln 2R z v R z C

Orbits in Axisymmetric Potentials (disk galaxies)

• cylindrical (R,,z) symmetry z-axis• stars in equatorial plane: same motions as in

spherically symmetric potential– non-closed rosette orbits

• stars moving out of plane– can be reduced to 2-D problem in (R,z)

– conservation of z-angular momentum, Lz

z

R

yx

R2=x2+y2

• Angular momentum about the z-axis is conserved, toque(rF=0) if no dependence on .

• Energy is also conserved (no time-dependence)

• Eliminating in the energy equation using conservation of angular momentum gives:

0)( 2222 Rdt

dRLZ

constzRzRR ),(2

1 2222

ER

JzRzR z

2

222

2),()(

2

1

Specific energy density in 3D

eff

Total Angular momentum almost conserved

• These orbits can be thought of as being planar with more or less fixed eccentricity.

• The approximate orbital planes have a fixed inclination to the z axis but they process about this axis.

• star picks up angular momentum as it goes towards the plane and returns it as it leaves.

Orbital energy• Energy of orbit is (per unit mass)

• effective potential is the gravitational potential energy plus the specific kinetic energy associated with motion in direction

• orbit bound within

2

eff

eff22

21

2

222

21

22221

E

zR

R

LzR

zRRE

z

• The angular momentum barrier for an orbit of energy E is given by

• The effective potential cannot be greater than the energy of the orbit.

• The equations of motion in the 2D meridional (RZ)plane then become: .

EzReff ),(

0

),(2222

zREzR eff

z

eff

eff

JR

zz

RR

2

• Thus, the 3D motion of a star in an axisymmetric potential (R,z) can be reduced to the motion of a star in a plane (Rz).

• This (non uniformly) rotating plane with cartesian coordinates (R,z) is often called the MERIDIONAL PLANE.

eff(R,z) is called the EFFECTIVE POTENTIAL.

• The orbits are bound between two radii (where the effective potential equals the total energy) and oscillates in the z direction.

• The minimum in eff occurs at the radius at which a circular orbit has angular momentum Lz.

• The value of eff at the minimum is the energy of this circular orbit.

R

eff

E

2

2

2R

J z

Rcir

• The effective potential is the sum of the gravitational potential energy of the orbiting star and the kinetic energy associated with its motion in the direction (rotation).

• Any difference between eff and E is simply kinetic energy of the motion in the (R,z) plane.

• Since the kinetic energy is non negative, the orbit is restricted to the area of the meridional plane satisfying .

• The curve bounding this area is called the ZERO VELOCITY CURVE since the orbit can only reach this curve if its velocity is instantaneously zero.

E - eff (R,z)>= 0

Nearly circular orbits: epicycles• In disk galaxies, many stars (disk stars) are

on nearly-circular orbits

• EoM:

• x=R-Rg

– expand in Taylor series about (x,z)=(0,0)

– then2/ 2/

0,at 0

;

2222

2

)0,(

2eff

2

212

)0,(

2eff

2

21

eff

effeff

effeff

zx

zz

xR

zRRzR

zz

RR

gg RR

g

• When the star is close to z=0 the effective potential can be expanded to give

22

2

2

1)0,(),( z

zz

zRzR eff

effeff

Zero, changes sign above/below z=0 equatorial plane. 2

.......2

1)0,(),( 22 zRzR effeff

zz 2

So, the orbit is oscillating in the z direction.

epicyclic approximation

• ignore all higher / cross terms:

• EoM: harmonic oscillators– epiclyclic frequency :– vertical frequency :

– with

2 2

2

2eff

222

2 4

( ,0)

R , and

( , )2

3;

g

z

z

gR

R z z

LR zR

L

R R

epicycles cont.

• using the circular frequency , given by

– so that

disk galaxy: ~ constant near centre– so ~ 2

~ declines with R, » slower than Keplerian R-3/2

» lower limit is ~

in general < 2

gR

z

z

R

RR

RRR

L

RR

R

L

RRR

22

224

22

4

2

)0,(

2

4

33

1)(

R

Vrot

Example:Oort’s constants near Sun

– where R0 is the galacto-centric distance

• then 2 = -4A(A-B) + 4(A-B)2 = -4B(A-B) = -4B0

• Obs. A = 14.5 km/s /kpc and B=-12 km/s /kpc

00

21

21 ;

RR RRB

RRA

2.03.12 0

0

BA

B

the sun makes 1.3 oscillations in the radial direction per azimuthal (2) orbit– epicyclic approximation not valid for z-motions

when |z|>300 pc

General Jeans Equations

• CBE of the phase space density f is eq. of 7 variables and hence generally difficult to solve

• Gain insights by taking moments of the CBE

• where integrate over all possible velocities– U=1, vj, vjvk

3

1

3 3 3

0

0

ii i i i

ii i i

f f fv

t x x v

f f fUd v v Ud v Ud v

t x x v

1st Jeans (continuity) equation

• define spatial density of stars n(x)

• and the mean stellar velocity v(x)

• then our zeroth moment equation becomes

n fd 3v

v i 1

nfvi d 3v

nt nv i xi

0

3rd Jeans Equation

similar to the Euler equation for a fluid flow:– last term of RHS represents pressure force

Pvvt

v

x

n

xn

x

vvn

t

vn

i

ij

ii

j

ij

1

2

JEANS EQUATION for oblate rotator

: a steady-state axisymmetrical system in which ASSUME ij

2 is isotropic and the only streaming motion is azimuthal rotation:

RR

v

R

zz

rot

22

2

)(1

)(1

• The velocity dispersions in this case are given by:

• If we know the forms of (R,z) and (R,z) then at any radius R we may integrate the Jeans equation in the z direction to obtain 2.

222

rotrot

2222

222222

isotropic sit' vfromapart since vvbut

2)(

since

),(

rot

rotrotrot

rotzr

vv

vvvvvv

vvvvzR

Obtaining 2

Inserting this into the jeans equation in the R

direction gives:

dzz

zRz

1

),(2

z

rot dzzR

R

RRv

2

20th Lec

• orbits

Applications of the Jeans Equations

• I. The mass density in the solar neighbourhood

• Using velocity and density distribution perpendicular to the Galactic disc– cylindrical coordinates.– Ignore R dependence

E.g.: Total Mass of spherical Milky WAY

• Motions of globular clusters and satellite galaxies around 100kpc of MW– Need n(r), vr

2, to find M(r), including dark halo

• Several attempts all suffer from problem of small numbers N ~ 15

• For the isotropic case, Little and Tremaine TOTAL mass of 2.4 (+1.3, -0,7) 1011 Msol

• 3 times the disc need DM

Power-law model of Milky Way

• Isotropic orbits: • Radial orbits• If we assume a power law for the density

distribution

– E.g. Flat rotation a=1, Self-grav gamma=2, Radial anis.

– E.g., Point mass a=0, Tracer gamma=3.5, Isotro

)(, rrMrn

0, v2 vr

2

1, v2 0

Grvvr /)(5.4M 22

Mass of the Milky Way: point-mass potential model

We find

2 2circ

2

2

2 2

1/

lnln2 1 2

ln ln

For =3.5, and isotropic tracer =0, we have

4.5

/ /

r

r

r

r

GM dr v v p

r dr

v r

d vd np

d r d r

p

M pv r G pv r G

Vertical Jeans equation

• Small z/R in the solar neighbourhood, R~8.5 kpc, |z|< 1kpc, R-dependence neglected.

• Hence, reduces to vertical hydrostatic eq.:

z

nvnz

z

2

mass density in solar neighbourhood

• Drop R, theta in Poisson’s equation in cylindrical coordinates:

GzRR

RRR

411

2

2

2

22

Gz

42

2

local mass density = 0

Finally

• all quantities on the LHS are, in principle, determinable from observations. RHS Known as the Oort limit.

• Uncertain due to double differentiation!

Gvnznz z 4/

1- 2

local mass density

• Don’t need to calculate for all stars– just a well defined population (ie G stars, BDs etc)

– test particles (don’t need all the mass to test potential)

• Procedure– determine the number density n, and the mean square vertical velocity, v z

2, the variance of the square of the velocity dispersion in the solar neighbourhood.

– need a reliable “tracer population” of stars• whose motions do not reflect formation• hence old population that has orbited Galaxy many times• ages > several x 109 years

• N.B. problems of double differentiation of the number density n derived from observations

• need a large sample of stars to obtain vz as f(z)

– ensure that vz is constant in time

– ie stars have forgotten initial motion

local mass density

• > 1000 stars required

• Oort : 0 = 0.15 Msol pc-3

• K dwarf stars (Kuijken and Gilmore 1989)– MNRAS 239, 651

• Dynamical mass density of 0 = 0.11 Msol pc-3

• also done with F stars (Knude 1994)

• Observed mass density of stars plus interstellar gas within a 20 pc radius is 0 = 0.10 Msol pc-3

• can get better estimate of surface density

• out to 700 pc ~ 90 Msol pc-2

• from rotation curve rot ~ 200 Msol pc-2

• Question 5:

).,(density star and

stars of mass totalcalculate ./1000V assume

term),(2nd stars and (1st term) halodark todue

,)1/)(1(

)2ln(5.0),( potentialIn 2/12222

0

2220

zR

skm

kpczRv

zRvzR

s

)0,1( unphysical haverotator isotropic Show

),1()0,1(stellar Calculate

(1kpc,0)?z)(R,equator on density halodark theisWhat

2

02

kpcv

dzz

zkpckpc

rot

ss

Helpful Math/Approximations(To be shown at AS4021 exam)

• Convenient Units

• Gravitational Constant

• Laplacian operator in various coordinates

• Phase Space Density f(x,v) relation with the mass in a small position cube and velocity cube

3dv3dx),(dM

)(spherical 2sin2r

2

sin2r

)(sin

2r

)2(

al)(cylindric 22-R2)(1-R

ar)(rectangul 222

1-sun

M2(km/s) kpc6104

1-sun

M2(km/s) pc3104

Gyr1

kpc1

1Myr

1pc 1km/s

vxf

rr

r

zRR

R

zyx

G

G

21th Lec: MOND

• orbits